>Logarithmic Equations>What Are Exponential Equations?

What Are Exponential Equations?

An exponential equation is an equation where $x$ is in the exponent. To get $x$ down from the exponent, you have to use logarithms. You are free to use either $\log$ or $\ln$ . When working with exponential and logarithmic equations it is important to remember the logarithmic and exponential rules! You solve these types of equations like this:

Rule

Exponential Equations

The same rules are in place for both

\ln

and

\log

. You are most likely to use

\ln

\begin{array}{l} a^{x} & = b \\ \ln a^{x} & = \ln b \end{array}

Do the logarithm on both sides to get $x$ down:

x \ln a = \ln b

Divide by $\ln a$ on both sides:

x = \frac{\ln b}{\ln a}, a, b > 0

Note! The answer will be the same whether you use $\ln$ or $\log$ !

Example 1

Solve the equation $1 0^{x} = 3000$

\begin{array}{l} 1 0^{x} = 3000 \ln 1 0^{x} & = \ln 3000 x & = \frac{\ln 3000}{\ln 10} x & \approx 3.477Example 2 Solve the equation 32 \cdot 2^{x} = 128 \begin{array}{l} 32 \cdot 2^{x} = 128 2^{x} & = 4 \ln 2^{x} & = \ln 4 x \ln 2 & = \ln 4 x & = \frac{\ln 4}{\ln 2} = \frac{\ln 2^{2}}{\ln 2} = \frac{2 \cdot \ln2}{\ln2} = 2Example 3 Solve the equation 5 \cdot 4^{x} = 45 \begin{array}{l} 5 \cdot 4^{x} = 45 4^{x} & = 9 \ln 4^{x} & = \ln 9 x \ln 4 & = \ln 9 x & = \frac{\ln 9}{\ln 4} = \frac{\ln 3^{2}}{\ln 2^{2}} = \frac{2 \ln 3}{2 \ln 2} = \frac{\ln 3}{\ln 2}Next entry What Are Logarithmic Equations?Go backHome Curriculum Stats LeagueAbout Us About House of Math Employees Career Media Lectures Blog Contact support@houseofmath.com Address House of Math 470 Ramona Street Palo Alto, CA 94301 USA FAQ Links WeTakeAction Math Magic for Ukraine Tutoring Junior Math Pen & Paper exercises Encyclopedia Language/Curriculum Terms Privacy Cookies Download our app{"props":{"pageProps":{"campaign":false,"articleBody":" An exponential equation is an equation where x is in the exponent. To get x down from the exponent, you have to use logarithms. You are free to use either log ⁡ or ln ⁡ . When working with exponential and logarithmic equations it is important to remember the logarithmic and exponential rules! You solve these types of equations like this: Rule Exponential Equations The same rules are in place for both ln ⁡ and log ⁡ . You are most likely to use ln ⁡ . ax = b ln ⁡ ax = ln ⁡ b Do the logarithm on both sides to get x down: x ln ⁡ a = ln ⁡ b Divide by ln ⁡ a on both sides: x = ln ⁡ b ln ⁡ a,a,b \u003e 0 Note! The answer will be the same whether you use ln ⁡ or log ⁡ ! Example 1 Solve the equation 10x = 3000 10x = 3000 ln ⁡ 10x = ln ⁡ 3000 x = ln ⁡ 3000 ln ⁡ 10 x ≈3.477 Example 2 Solve the equation 32 ⋅ 2x = 128 32 ⋅ 2x = 128| ÷ 32 2x = 4 ln ⁡ 2x = ln ⁡ 4 x ln ⁡ 2 = ln ⁡ 4 x = ln ⁡ 4 ln ⁡ 2 = ln ⁡ 22 ln ⁡ 2 = 2 ⋅ ln ⁡ 2 ln ⁡ 2 = 2 Example 3 Solve the equation 5 ⋅ 4x = 45 5 ⋅ 4x = 45| ÷ 5 4x = 9 ln ⁡ 4x = ln ⁡ 9 x ln ⁡ 4 = ln ⁡ 9| ÷ ln ⁡ 4 x = ln ⁡ 9 ln ⁡ 4 = ln ⁡ 32 ln ⁡ 22 = 2 ln ⁡ 3 2 ln ⁡ 2 = ln ⁡ 3 ln ⁡ 2 ","topic":{"breadcrumbs":[{"_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c","h1":"Encyclopedia","path":"/encyclopedia"},{"_key":"e9327e4b-42fe-571b-bcb9-83fdb63d5f21","h1":"Algebra","path":"/encyclopedia/algebra"},{"_key":"5746e0ab-24ae-5841-af0f-f1b35cfab6c9","h1":"Logarithms","path":"/encyclopedia/algebra/logarithms"},{"_key":"db5be093-ee70-5f75-bdd2-f3d4e77d9fda","h1":"Logarithmic Equations","path":"/encyclopedia/algebra/logarithms/logarithmic-equations"},{"_key":"ccac7cf1-c080-5188-afab-eeebf16de339","h1":"What Are Exponential Equations?","path":"/encyclopedia/algebra/logarithms/logarithmic-equations/what-are-exponential-equations"}],"cssUrl":"/8c2ce5fa-9e80-4618-888c-fb8e70afc5d1-en-1.css","h1":"What Are Exponential Equations?","htmlContent":{"html":" \u003c/p\u003e\u003cp class=\"indent\" \u003e     An \u003cspan  class=\"ec-lmri-12x-x-144\"\u003eexponential equation \u003c/span\u003eis an equation where \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/math\u003e is in the exponent. To get \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/math\u003e down from the exponent, you have to use \u003cspan  class=\"ec-lmri-12x-x-144\"\u003elogarithms. \u003c/span\u003eYou are free to use either \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003c/math\u003e or \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003c/math type=\"punct\"\u003e. \u003c/p\u003e\u003cp class=\"indent\" \u003e     When working with exponential and \u003ca href=\"/encyclopedia/algebra/logarithms/logarithmic-equations\"\u003elogarithmic equations\u003c/a\u003e it is important to remember the \u003ca href=\"/encyclopedia/algebra/logarithms/about-logarithms/what-are-the-logarithm-rules\"\u003elogarithmic and exponential rules\u003c/a\u003e! You solve these types of equations like this: \u003c/p\u003e                                                                                                                                                                                                                                               \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-693\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Rule\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"rule-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eExponential\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eEquations\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eThe same rules are in place for both \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003c/math\u003e. You are most likely to use \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003c/math type=\"punct\"\u003e. \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                                                \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                        \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-label\"\u003e                                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                                     \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003eDo the logarithm on both sides to get \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/math\u003e down: \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                                \u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e \u003cp class=\"noindent\" \u003eDivide by \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e on both sides: \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                            \u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\" class=\"quad\"/\u003e\u003cmi  \u003ea\u003c/mi\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmi  \u003eb\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e \u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eNote!\u003c/span\u003e\u003c/b\u003e The answer will be the same whether you use \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003c/math\u003e or \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003c/math\u003e! \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                              \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-694\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the equation \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                                   \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e                                   \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e                                   \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e   \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                  \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e                                   \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e≈\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                          \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                                          \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-695\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the equation \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                      \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmspace width=\"1em\" class=\"quad\"/\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                          \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e   \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmtext class=\"cancel\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mtext\u003e\u003c/mrow\u003e    \u003cmrow  \u003e\u003cmtext class=\"cancel\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mfrac\u003e   \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                      \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-696\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 3\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the equation \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                      \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmspace width=\"2em\" class=\"qquad\"/\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                        \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"1em\" class=\"quad\"/\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                           \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                      \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                      \u003cp class=\"indent\" \u003e     "},"image":null,"isRootPath":false,"key":"content/en/encyclopedia/02 Algebra/03 Logarithms/03 Logarithmic 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You heard right - without doing math!","subtitle.teacher":"It is built to optimize student learning, so the teacher panel is not complete yet."},"math-practice-challenges":{"title":{"student":"My math practice could be more challenging","parent":"I see untapped potential in my children","teacher":"I see untapped potential in many of my students","other":"My math practice could be more challenging"}},"math-routine":{"title":{"student":"Mastering math makes me feel smart","parent":"My support makes a real difference in my children's math progress","teacher":"I have time to make all my students feel seen","other":"Mastering math makes me feel smart"}},"succeed-in-something":{"title.student":"When you get stuck, what do you do first?","title.teacher":"How do you usually celebrate student accomplishments?","title.other":"When you succeed in something, you'd rather...","title.parent":"How do you celebrate your children's accomplishments?","subtitle":"","student":{"think-of-what-could-be-done-better":{"title":"Give up pretty fast"},"celebrate-the-win":{"title":"Search for help"}},"parent":{"think-of-what-could-be-done-better":{"title":"Think about the next step"},"celebrate-the-win":{"title":"Celebrate the win"}},"teacher":{"think-of-what-could-be-done-better":{"title":"Think about the next step"},"celebrate-the-win":{"title":"Celebrate the win"}}},"time-you-commit":{"title.student":"How much time can you commit?","title.teacher":"Do you feel you catch learning gaps in time?","title.other":"How much time can you commit?","title.parent":"Rewards help motivate my children to practice math","subtitle":"","student":{"5-min":{"title":"5 min/day","subtitle":"quick daily practice"},"15-min":{"title":"15 min/day","subtitle":"steady progress"},"30-min":{"title":"30 min/day","subtitle":"Fast improvement"},"60-min":{"title":"60+ min/day","subtitle":"Maximum results"}},"parent":{"5-min":{"title":"Works like a charm","subtitle":"Finally self-going"},"15-min":{"title":"Increases stamina","subtitle":"They do more math"},"30-min":{"title":"Not sure","subtitle":"Does it have an effect?"},"60-min":{"title":"It is noise","subtitle":"They lose focus"}},"teacher":{"5-min":{"title":"Yes, full control","subtitle":"Impressive!"},"15-min":{"title":"Most of the time","subtitle":"We'll take you the last stretch"},"30-min":{"title":"Now and then","subtitle":"We'll help you see all"},"60-min":{"title":"Not really","subtitle":"No worries, we'll fix that"}}},"find-challenging-dragon":{"title":{"parent":{"1":"Helping with math feels difficult and uncertain","2":"You can help sometimes, but it doesn't always feel easy","3":"You generally feel comfortable helping your children with math","4":"Your confidence is a real advantage for your children"},"student":{"1":"Your future isn't decided — still much time to shape it into what you want","2":"Growth often starts with small steps and support","3":"Progress comes from showing up and trying – skills for life!","4":"Effort opens doors to math and a bright future"},"teacher":{"1":"One-size-fits-all homework often falls short","2":"Designing homework for mixed levels is hard","3":"Homework mostly works, but edge cases remain","4":"You already differentiate effectively"},"other":{"1":"Your future isn't decided — still much time to shape it into what you want","2":"Growth often starts with small steps and support","3":"Progress comes from showing up and trying – skills for life!","4":"Effort opens doors to math and a bright future"}},"subtitle":{"parent":{"1":"We'll guide you step by step so you know exactly how to support your children — even if math wasn't your strong subject.","2":"We'll strengthen your understanding and give you clear guidance so you feel more confident when your children need support.","3":"We'll help you stay aligned with what they need most to keep their progress — and grades — moving up.","4":"We'll help you use that strength strategically to support top performance and strong test results."},"student":{"1":"Our founder flunked math in high school and became a mathematician now helping millions master math. We know how to help you. Let's go!","2":"Even a little effort can unlock understanding and new possibilities over time. We are here to get you going and guide you on your way.","3":"Steady effort leads to improvement and prepares you for bigger challenges ahead. We'll take you even one step further.","4":"Working hard builds skills, confidence, and habits that create opportunities in school and in life. We are here to make your journey fast, easy and fun!"},"teacher":{"1":"We'll map individual gaps and generate personalized practice that supports every learner so they progress between every class","2":"We'll identify where students diverge and create targeted paths that meet them where they are. That way all students progress between classes.","3":"We'll help you spot where adjustments can unlock better outcomes for more students. That way all students progress between classes.","4":"You stay in control, with clear insight into how to support each student and maintain progress between classes."},"other":{"1":"Our founder flunked math in high school and became a mathematician now helping millions master math. We know how to help you. Let's go!","2":"Even a little effort can unlock understanding and new possibilities over time. We are her to get you going and guide you on your way.","3":"Steady effort leads to improvement and prepares you for bigger challenges ahead. We'll take you even one step further.","4":"Working hard builds skills, confidence, and habits that create opportunities in school and in life. We are here to make your journey fast, easy and fun!"}}},"personal-math-recipe":{"title.student":{"1":"Your personal learning recipe is 72% ready!","2":"Your personal learning recipe is 72% ready!","3":"Your personal learning recipe is 72% ready!","4":"Your personal learning recipe is 72% ready!"},"title.other":{"1":"Your personal learning recipe is 72% ready!","2":"Your personal learning recipe is 72% ready!","3":"Your personal learning recipe is 72% ready!","4":"Your personal learning recipe is 72% ready!"},"title.parent":{"1":"Your learning analysis is 72% ready!","2":"Your learning analysis is 72% ready!","3":"Your learning analysis is 72% ready!","4":"Your learning analysis is 72% ready!"},"title.teacher":{"1":"Your learning analysis is 72% ready!","2":"Your learning analysis is 72% ready!","3":"Your learning analysis is 72% ready!","4":"Your learning analysis is 72% ready!"},"subtitle.parent":{"1":"Just a little more, and you are ready to give your children a new math start","2":"Just a little more, and you are ready to give your children a new math start","3":"Just a little more, and you are ready to give your children a new math start","4":"Just a little more, and you are ready to give your children a new math start"},"subtitle.teacher":{"1":"Just a little more, and you are ready to give your students a new math start","2":"Just a little more, and you are ready to give your students a new math start","3":"Just a little more, and you are ready to give your students a new math start","4":"Just a little more, and you are ready to give your students a new math start"},"subtitle.other":{"1":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","2":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","3":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","4":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe."},"subtitle.student":{"1":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","2":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","3":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","4":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe."}},"friendly-push-dragon":{"title":{"student":{"1":"You rarely need external encouragement","2":"An occasional nudge can be helpful","3":"A friendly push is a powerful motivator","4":"Support and encouragement make a big difference for you"},"other":{"1":"You rarely need external encouragement","2":"An occasional nudge can be helpful","3":"A friendly push is a powerful motivator","4":"Support and encouragement make a big difference for you"},"parent":{"1":"Stressful and hard to manage","2":"Often challenging","3":"Mostly manageable","4":"Smooth and stress-free"},"teacher":{"1":"You're not alone! It can be hard to know if the basics are truly sticking","2":"Foundations can look solid until the content ramps up","3":"You're in a good place","4":"It sounds like your students are on a strong track"}},"subtitle":{"student":{"1":"You're highly self-driven. You're confident in staying motivated and moving forward on your own. Excellent trait to have to master math. But being seen is always nice, so we'll be here to cheer you on.","2":"You're generally self-motivated. You usually stay on track without much outside support. We'll be your little friendly reminder to keep going steadily.","3":"We'll tailor practice, explanations, and recommendations to help you keep progressing toward mastery.","4":"We'll proactively guide and motivate you with timely nudges to help you stay focused and make steady progress with your math."},"other":{"1":"You're highly self-driven. You're confident in staying motivated and moving forward on your own. Excellent trait to have to master math. But being seen is always nice, so we'll be here to cheer you on.","2":"You're generally self-motivated. You usually stay on track without much outside support. We'll be your little friendly reminder to keep going steadily.","3":"We'll tailor practice, explanations, and recommendations to help you keep progressing toward mastery.","4":"We'll proactively guide and motivate you with timely nudges to help you stay focused and make steady progress with your math."},"parent":{"1":"Homework at home is usually stressful and difficult for our family to handle.","2":"Homework time can be tough, and my children often get stuck or frustrated.","3":"Most days it goes fine, but sometimes my children need extra time or a bit of help from me.","4":"Homework time at home is usually calm, and my children can finish without much stress."},"teacher":{"1":"A structured baseline check can pinpoint missing prerequisites and give you a clear plan for strengthening their foundation step by step.","2":"A targeted diagnostic can quickly show which core skills need reinforcing so you can focus your time where it matters most","3":"A short skills check can validate the foundation and reveal any small gaps before they become barriers in higher-level topics","4":"A quick mastery snapshot can help confirm consistency across the class and highlight who's ready for stretch challenges"}}},"personal-math-recipe-2":{"title.student":{"1":"Your personal learning recipe is 94% ready!","2":"Your personal learning recipe is 94% ready!","3":"Your personal learning recipe is 94% ready!","4":"Your personal learning recipe is 94% ready!"},"title.other":{"1":"Your personal learning recipe is 94% ready!","2":"Your personal learning recipe is 94% ready!","3":"Your personal learning recipe is 94% ready!","4":"Your personal learning recipe is 94% ready!"},"title.parent":{"1":"Your learning analysis is 94% ready!","2":"Your learning analysis is 94% ready!","3":"Your learning analysis is 94% ready!","4":"Your learning analysis is 94% ready!"},"title.teacher":{"1":"Your learning analysis is 94% ready!","2":"Your learning analysis is 94% ready!","3":"Your learning analysis is 94% ready!","4":"Your learning analysis is 94% ready!"},"subtitle.student":{"1":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","2":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","3":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","4":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps."},"subtitle.other":{"1":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","2":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","3":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","4":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps."},"subtitle.parent":{"1":"You are nearly there now!","2":"You are nearly there now!","3":"You are nearly there now!","4":"You are nearly there now!"},"subtitle.teacher":{"1":"You are nearly there now!","2":"You are nearly there now!","3":"You are nearly there now!","4":"You are nearly there now!"}},"succeed-in-something-dragon":{"title":{"student":{"1":"Getting stuck can feel frustrating and overwhelming","2":"Not giving up is a winning strategy in math and life"},"other":{"1":"Getting stuck can feel frustrating and overwhelming","2":"Not giving up is a winning strategy in math and life"},"parent":{"1":"You're a math improver","2":"You balance progress and celebration"},"teacher":{"1":"You're forward-looking by nature","2":"You balance progress and celebration"}},"subtitle":{"parent":{"1":"You like refining your skills and spotting what to fix next. We'll use that mindset to provide targeted feedback and practice that tightens the small gaps fast.","2":"You notice what works and what needs improvement. We'll tailor the learning experience with steady challenges and clear milestones, so your children will keep moving forward and feel the wins."},"student":{"1":"It's normal to feel overwhelmed sometimes. We'll support you through tough moments to turn struggle into learning breakthroughs.","2":"Asking questions and using available resources builds understanding and confidence. We'll be your No. 1 resource in these moments."},"teacher":{"1":"You like refining their skills and spotting what they need fixing next. We'll use that mindset to provide targeted feedback and practice that tightens the small gaps fast.","2":"You notice what works and what needs improvement. We'll tailor the learning experience with steady challenges and clear milestones, so your students will keep moving forward and feel the wins."},"other":{"1":"It's normal to feel overwhelmed sometimes. We'll support you through tough moments to turn struggle into learning breakthroughs.","2":"Asking questions and using available resources builds understanding and confidence. We'll be your No. 1 resource in these moments."}}},"like-math-dragon":{"title":{"parent":{"1":"Math feels challenging and not very enjoyable for you","2":"Math is not your favorite subject, but you're willing to work with it ","3":"You generally like math and enjoy working with it","4":"Your enthusiasm is key to achieve success"},"student":{"1":"Math feels challenging and not very enjoyable for you","2":"Math is not your favorite subject, but you're willing to work with it","3":"You generally like math and enjoy working with it","4":"Your enthusiasm is key to achieve success"},"teacher":{"1":"You handle mixed-ability classrooms with confidence.","2":"You've found ways to manage differences in ability","3":"Different ability levels is a challenge in some lessons","4":"Teaching wide ability ranges can feel overwhelming"},"other":{"1":"Math feels challenging and not very enjoyable for you","2":"Math is not your favorite subject, but you're willing to work with it","3":"You generally like math and enjoy working with it","4":"Your enthusiasm is key to achieve success"}},"subtitle":{"parent":{"1":"Don't worry! 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We help you build real understanding to keep your momentum going."}}},"feel-math-skills-dragon":{"title":{"parent":{"1":"Motivation is a real challenge right now","2":"Motivation comes and goes","3":"Your children are generally motivated.","4":"Your children are highly motivated"},"student":{"1":"We'll make you feel less nervous and more confident before your math tests","2":"We'll make you feel less nervous before your math tests","3":"Math tests is a moment of  happiness for you","4":"You love taking math tests and we understand why!"},"teacher":{"1":"Motivation is often a challenge","2":"Motivation comes and goes","3":"Most students are generally motivated","4":"Your students are highly motivated to learn math"},"other":{"1":"We'll make you feel less nervous and more confident before your math tests","2":"We'll make you feel less nervous before your math tests","3":"Math tests is a moment of  happiness for you","4":"You love taking math tests and we understand why!"}},"subtitle":{"parent":{"1":"We'll help make progress visible and rewarding, so your children stay engaged and confident","2":"We'll support you with strategies that keep your children focused and moving forward — especially when things get hard.","3":"We'll help maintain that momentum with clear goals and progress tracking","4":"We'll help channel that drive into consistent improvement and strong test performance."},"student":{"1":"That's okay. 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And we will get you on the right track."},"teacher":{"1":"We'll help you focus on practical strategies that make progress possible, for all students","2":"We'll support you in using our platform to expand your reach without adding pressure","3":"We'll help you turn that belief into consistent, classroom-wide impact","4":"We'll help you put them to work so all learners can succeed in your classroom"},"other":{"1":"That's a powerful mindset. You believe math is learnable, and with effort and practice you can keep improving. We'll help you strengthen your foundation, close any gaps, and move forward with confidence.","2":"Even if it feels challenging at times, we'll help you strengthen confidence as your skills grow.","3":"We'll help you build understanding step by step and show that math skills can be learned.","4":"Millions of students feel like this BUT it is a misconception. The reason for this feeling is that you are trying ot learn math while having too many knowledge gaps. 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And we will get you on the right track."}}},"math-plan-method":{"title":{"parent":"House of Math is based on neuroscience, psychology, pedagogy and didactical methods","student":"Your learning recipe is based on neuroscience, psychology, pedagogy and didactical methods","teacher":"House of Math is based on neuroscience, psychology, pedagogy and didactical methods","other":"Your learning recipe is based on neuroscience, psychology, pedagogy and didactical methods"}},"stuck-to-got":{"title":{"parent":"Turn \u003citalic\u003eI'm stuck\u003c/italic\u003e into \u003citalic\u003eI've got this\u003c/italic\u003e","student":"Turn \u003citalic\u003eI'm stuck\u003c/italic\u003e into \u003citalic\u003eI've got this\u003c/italic\u003e","teacher":"Turn reaching \u003citalic\u003e“some students”\u003c/italic\u003e into reaching \u003citalic\u003e“all students”\u003c/italic\u003e","other":"Turn \u003citalic\u003eI'm stuck\u003c/italic\u003e into \u003citalic\u003eI've got this\u003c/italic\u003e"},"subtitle":{"parent":"Based on your answers, we've tailored a personal learning protocol.","student":"Based on your answers, we've tailored a personal learning protocol.","teacher":"Based on your answers, we would love to have you sign up and join our free teacher trial, where you can experience how the platform  works for a student","other":"Based on your answers, we've tailored a personal learning protocol."},"archaic-learning":{"title":"ARCHAIC LEARNING","subtitle":"Too general ⏳","dialog":"Too much at once"},"smart-practice":{"title":"SMART PRACTICE","subtitle":"Targeted practice ⚡","dialog":"few minutes a day"}},"email-us":{"title":"Join Our Teacher Beta Program","subtitle":"Get a free Class Learning Analysis and explore our premium teaching tools","emailDescription":"Enter your email — we'll send you a free Learning Analysis based on your answers","emailLabel":"Educational institution email address","emailPlaceholder":"your@school.edu","submitButton":"Get my free Analysis","betaTitle":"TEACHER BETA PROGRAM","betaDescription":"Want full access to our premium teaching tools? Send us an email to join the beta.","emailUsButton":"E-mail us","emailSentSuccess":"Email sent successfully","helperText":"Use your educational institution email address","signUp":"Or Sign up for a free trial","message":"I want to receive deals and communication from House of Math"}},"paymentOnboarding":{"title":"Payment","heading":{"student":"It's time to get the math grade you deserve","parent":"Give your children the support they deserve to bridge the gaps","parentV1":"Your Family's Math Success Starts Here","teacher":"Unlock the math tools to transform your students","other":"It's time to get the math grade you deserve"},"mostPopular":"MOST POPULAR","monthly":"Monthly","yearly":"Yearly","annually":"Annually","pricePerDay":"\u003cstrong\u003e{{pricePerDay}}\u003c/strong\u003e per day","pricePerMonth":"\u003cstrong\u003e{{pricePerMonth}}\u003c/strong\u003e per month","savePercentage":"\u003corange\u003eSAVE {{savedPercentage}}%\u003c/orange\u003e","zeroToday":"$0 today","startFreeTrial":"{{days}} Day Free Trial","bestValueMostPopular":"Best value • Most popular choice","priceMonthly":"\u003cstrong\u003e{{price}}\u003c/strong\u003e/mo","priceYearly":"\u003cstrong\u003e{{price}}\u003c/strong\u003e/yr","billedAnnually":"{{price}}/billed annually","yearlyBilledNote":"*Billed as one annual payment of {{price}}. 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Students in Texas, Virginia, Alaska, and Nebraska can also achieve strong learning outcomes with House of Math, but should use the our AI Oracle to ensure they cover all required topics. The platform supports all K–12 math globally, as well as selected university-level subjects.","cancelAnytime":"Yes. You can cancel whenever you want, and you'll keep access until the end of your current billing period. No hidden fees.","gradeLevelsCover":"We cover the full K–12 U.S. curriculum. Lessons are reviewed and verified by expert educators.","freeTrial":"Yes, we offer a 7 day free trial. Cancel anytime during the trial and you won't be charged."}}}},"getStartedV2":{"freeTakes90Seconds":"Free · Takes 90 seconds","understandChildMath":"Let's understand what's happening with your child’s math","answerQuestions":"Answer a few quick questions about what you're seeing at home. 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The real cause is usually a few missing foundational blocks that quietly make every lesson feel scary. House of Math finds exactly which blocks need rebuilding.","ES_EMO_NEW":"Seeing your child shrink away from math is heartbreaking, especially this young. When {{breakdown}} starts to feel scary, it often traces back to a few foundational blocks that never fully solidified — not to lack of effort. House of Math finds exactly which blocks are missing and rebuilds them.","ES_ACA_EXP":"You've already invested in outside support — yet progress in {{breakdown}} hasn't matched the effort. More hours rarely fixes this; finding the specific foundational blocks your child skipped does. House of Math identifies exactly those gaps and rebuilds them.","ES_ACA_NEW":"Early math foundations shape everything that comes later, so wanting your child to do well now is exactly the right instinct. 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Please try again shortly.","additionalMembers":"Additional Members","billingMonthly":"Monthly","billingYearly":"Yearly · Save {{percent}}%","planMonthly":"My monthly plan","planYearly":"My yearly plan","perMonth":"{{price}} / month","perYear":"{{price}} / year","additionalMemberLine":"Additional member","amountToPay":"Amount to pay","haveDiscountCode":"Have a discount code? ","enterItHere":"Enter it here","reactivatePlan":"Reactivate subscription","discountCode":"Discount code","discountCodePlaceholder":"Code","applyDiscount":"Apply","dismissDiscount":"Remove discount code","discountApplied":"Discount applied","maxLicenses":"Maximum 10 licenses","partnerCoupon":{"appliedLabel":"APPLIED PARTNER DISCOUNT","dnbTitle":"DNB Partner","knowitTitle":"Knowit Partner","dnbAppliedVia":"Applied via DnB partnership","knowitAppliedVia":"Applied via Knowit partnership","badgeLabel":"Partner pricing","codeLabel":"Partner code","description":"Your partner discount is already applied. You can't combine partner pricing with other coupons."}},"FAQ":{"title":"FAQ","question":{"whatIsMathAccess":"What is a math access?","doAdminNeedMathAccess":"Do admin need admin math access?","childEmailRequirement":"Do my child need to have their own email to access House of Math?","subscriptionTransfer":"What happens if I add a user who already has a subscription to the Family Account?"},"answer":{"whatIsMathAccess":"Math access gives the user access to the entire House of Math platform, including exercises, videos, games, and statistics.","doAdminNeedMathAccess":"No, only if the admin also wants to use the learning content themselves. Without math access, the admin can still use the parent panel to follow the child’s progress in mathematics.","childEmailRequirement":"No, it’s not required. 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