Hvordan løse logaritmelikninger

Logaritmelikninger er likninger der $x$ er en del av argumentet til logaritmen. For å løse ut $x$ må du derfor bruke logaritmesetningene eller potenser til å bli kvitt logaritmen. Når du bruker $\log$ bruker du $10$ som grunntal og når du bruker $\ln$ bruker du $e$ som grunntall.

Teori

Logaritmelikninger

I logaritmelikninger blir du kvitt logaritmen ved å opphøye begge sider av likningen med 10 eller $e$ som grunntall. Du bruker 10 når du har en Briggsk logaritme og $e$ når du har en naturlig logaritme.

\begin{array}{l} \lg x & = c & \ln x & = c \\ 1 0^{\lg x} & = 1 0^{c} & e^{\ln x} & = e^{c} \\ x & = 1 0^{c} & x & = e^{c} \end{array}

NB! Du kan bare ta logaritmen til positive tall.

Eksempel 1

Løs logaritmelikningen $\ln x = 4$

\begin{array}{l} ln x = 4 e^{\ln x} & = e^{4} x & = e^{4} x & \approx 54, 6Eksempel 2 Løs likningen \frac{1}{4} \lg 2 x = 8 \begin{array}{l} 14 \ln 2 x = 8 | \cdot 4 \ln 2 x & = 32 e^{\ln 2 x} & = e^{32} 2 x & = e^{32} & | : 2 x & = \frac{e^{32}}{2} \approx 3, 95 \cdot 1 0^{13}Eksempel 3 Løs likningen 5 \ln (x + 2) = 15 \begin{array}{l} 5 \lnVil du vite mer? Registrer deg Det er gratis!Forrige oppslag Hvordan løse eksponentiallikninger Neste oppslag Hvordan løse eksponentialulikheterTilbakeHjem All matte Spill Statistikk ChatOm oss Om House of Math Om ansatte Karriere Media Foredrag Blogg Kontakt support@houseofmath.com Booking av privatundervisning +47 22 150 300 Adresse Bygdøy allé 23
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For å løse ut \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/math\u003e må du derfor bruke \u003ca href=\"/no/encyclopedia/algebra/logaritmer/om-logaritmer/hva-er-regnereglene-for-logaritmer\"\u003elogaritmesetningene\u003c/a\u003e eller \u003ca href=\"/no/encyclopedia/tall-og-maaling/regning/potenser-og-rotter/potensregler\"\u003epotenser\u003c/a\u003e til å bli kvitt logaritmen. Når du bruker \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 bruker du \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/math\u003e som grunntal og når du bruker \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 bruker du \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ee\u003c/mi\u003e\u003c/math\u003e som grunntall. \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooinfo\" id=\"tcolobox-637\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Teori\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"info-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eLogaritmelikninger\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eI logaritmelikninger blir du kvitt logaritmen ved å opphøye begge sider av likningen med 10 eller \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ee\u003c/mi\u003e\u003c/math\u003e som grunntall. Du bruker 10 når du har en Briggsk logaritme og \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ee\u003c/mi\u003e\u003c/math\u003e når du har en naturlig logaritme. \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\u003cmi class=\"qopname\"\u003e lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmi  \u003ec\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                         \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e 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\u003cmn\u003e8\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                             \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmfrac\u003e\u003cmrow  \u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e \u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e8\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                              \u003cmtd  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ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\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-odd\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\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\u003cmi  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\u003e\u003c/mtext\u003e\u003c/mstyle\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmi  \u003e,\u003c/mi\u003e\u003cmn\u003e1\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                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     "},"breadcrumbs":[{"path":"/encyclopedia","h1":"Oppslagsverk","_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c"},{"path":"/encyclopedia/algebra","h1":"Algebra","_key":"e9327e4b-42fe-571b-bcb9-83fdb63d5f21"},{"path":"/encyclopedia/algebra/logaritmer","h1":"Logaritmer","_key":"59f2f1f4-5fb4-5961-b91c-f612c8849002"},{"path":"/encyclopedia/algebra/logaritmer/logaritmelikninger","h1":"Logaritmelikninger","_key":"d6b72b45-d8fa-5d3f-946b-afd66760b788"},{"h1":"Hvordan løse logaritmelikninger","_key":"f1343c4c-c62d-531e-ab59-ba3cf50b36e4","path":"/encyclopedia/algebra/logaritmer/logaritmelikninger/hvordan-lose-logaritmelikninger"}],"taskReference":null,"type":"encyclopedia","h1":"Hvordan løse 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