>Logaritmelikninger>Hvordan løse logaritmelikninger

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 Progresjon LigaOm 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. 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                                                                                                    \u003cp class=\"indent\" \u003e     Eksempel 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\"\u003eLøs\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003elikningen\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\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 lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e 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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  \u003ee\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003eln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e2\u003c/mn\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 \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ee\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \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\" 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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\u003c/mtable\u003e\u003c/math\u003e \u003c/b\u003e                                                                                                                                        \u003c/div\u003e  \u003c/div\u003e                                                                                                                           \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-1447\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Eksempel 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\"\u003eLøs likningen \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\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003ex\u003c/mi\u003e 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                                             \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 \u003cmo  class=\"MathClass-bin\"\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 \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ee\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   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Ønsker du å bytte språk på plattformen til norsk også?","changeCurriculum":"Endre bare pensum","changeCurriculumAndLanguage":"Endre språk på plattform og pensum","goal":"Mål {{code}}","viewGrade":"Se pensum","ChangeGrade":"Endre klassetrinn","confirm":"Bekreft valget"},"gemShop":{"buyNow":"Kjøp","comingSoon":"Kommer snart","itemTypes":{"chest":"Overkropp","head":"Hode","neck":"Nakke","hats":"Mine hodeplagg","jewelry":"Mine smykker","clothes":"Mine klær","special":"Unik"},"menu":{"pet":"Kjæledyr","featured":"Utvalgte"},"noGems":"Trenger flere krystaller","owned":"Kjøpt","store":"Butikk","tier":"Nivå","upgrade":"Oppgradering","upgradeTypes":{"productionRelativeIncrease":"+{{value}}% krystallproduksjon","productionAbsoluteIncrease":"+{{value}} krystallproduksjon","tickRateIncrease":"Øk farten på timeren med {{value}}x","energyCapacityIncrease":"Kudoskapasitet +{{value}}"},"unlockRank":"Dragenivå {{rank}}","padlockAlt":"Hengelås","rankUp":"Øk dragens nivå for å låse opp denne varen.","allItems":"Alle"},"homebase":{"mainView":"Base","dragonView":"Drage","storeView":"Butikk","cavern":"Mørk grotte","checkBaseForQuests":"Sjekk ut læringsstien din","chestText":"Dette er din base. 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Bruk tiden til å samle mer Matteenergi fra kistene i Drillheftene!","DoNotHaveEnoughMnO":"Du har ikke mer Kudos! Gjør flere matteoppdrag for å samle flere!","DragonFull":"Dragen din er mett! Vent en stund før du gir den mer Kudos.","DragonTrickCooldown":"Dragen din er sliten etter å ha trent på det nye trikset. Prøv igjen senere!","DragonTrickNotWilling":"Dragen din vil ikke trene mer før den er mett og fornøyd!","DragonTutorial1":"Sørg for å holde dragen din sunn og god ved å gi den Kudos!","DragonTutorial2":"Ryktene sier at dragene kan lære seg nye triks hvis du holder de mette en stund!","DragonTutorial3":"Drager trenger også kos! Sørg for å kose med den jevnlig!","DragonTutorial4":"Kult! Dragen din har det skikkelig bra! Sørg for å gi den oppmerksomhet hver dag!","EggHasBeenChosen":"Godt valg! La oss plassere egget trygt i basen din. Kan vi få det til å klekke?","EggHungry1":"Hmm... 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