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}Want to know more? Sign Up It's free!Next entry What Are Logarithmic Equations?Go backHome All Math Games Statistics ChatAbout Us About House of Math Employees Career Media Lectures Blog Contact support@houseofmath.com Booking of private tutoring +47 22 150 300 Address Bygdøy allé 23
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            Norway FAQ Links WeTakeAction Math Magic for Ukraine Tutoring Video Crash Courses Junior Math Math Essentials Encyclopedia Language Norsk Українська Terms Privacy{"props":{"pageProps":{"campaign":false,"topic":{"breadcrumbs":[{"path":"/encyclopedia","h1":"Encyclopedia","_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c"},{"path":"/encyclopedia/algebra","h1":"Algebra","_key":"e9327e4b-42fe-571b-bcb9-83fdb63d5f21"},{"path":"/encyclopedia/algebra/logarithms","h1":"Logarithms","_key":"5746e0ab-24ae-5841-af0f-f1b35cfab6c9"},{"path":"/encyclopedia/algebra/logarithms/logarithmic-equations","h1":"Logarithmic Equations","_key":"db5be093-ee70-5f75-bdd2-f3d4e77d9fda"},{"path":"/encyclopedia/algebra/logarithms/logarithmic-equations/what-are-exponential-equations","h1":"What Are Exponential Equations?","_key":"ccac7cf1-c080-5188-afab-eeebf16de339"}],"path":"/encyclopedia/algebra/logarithms/logarithmic-equations/what-are-exponential-equations","title":"What Are Exponential Equations?","type":"encyclopedia","language":"en","metaDescription":"Learn how to use the rules for logarithms to solve exponential equations. Logarithms will allow you to get a variable down from an exponent.","cssUrl":"/80bd8b3e-74e6-41e8-a473-0237fd221bbf-en-1.css","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                                                                                                                           \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/b\u003e                                                                                                                     \u003c/div\u003e  \u003c/div\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/b\u003e                                                                                               \u003c/div\u003e  \u003c/div\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/b\u003e                                                                                               \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     "},"taskReference":null,"key":"content/en/encyclopedia/02 Algebra/03 Logarithms/03 Logarithmic Equations/301ExponentialEquations.tex","previewUrl":"https://images.houseofmath.com/eyJrZXkiOiJvdmVybGVhZi1hc3NldHMvcHJldmlldy8zMDFleHBvbmVudGlhbGVxdWF0aW9ucy5wbmciLCJidWNrZXQiOiJob20tZnJvbnRlbmQtYXNzZXRzIiwidmVyc2lvbiI6M30=","locale":"en","isRootPath":false,"otherLocales":[],"h1":"What Are Exponential Equations?","image":null,"siblings":[{"h1":"What Are Exponential Equations?","path":"/encyclopedia/algebra/logarithms/logarithmic-equations/what-are-exponential-equations"},{"h1":"What Are Logarithmic Equations?","path":"/encyclopedia/algebra/logarithms/logarithmic-equations/what-are-logarithmic-equations"},{"h1":"How to Solve Exponential Inequalities","path":"/encyclopedia/algebra/logarithms/logarithmic-equations/how-to-solve-exponential-inequalities"},{"h1":"How to Solve Logarithmic 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