>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 Pensum Spill Stats 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. 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-1444\"\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. 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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                           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Dette har ført til at prestasjonene mine har blitt bedre og jeg har fått bedre karakterer.","reviewThorbjorn":"Jeg sendte datteren min til House of Math da hun begynte på ungdomsskolen, ikke fordi hun slet med matte, men fordi hun trengte motivajon. Tenåringshodet hennes trengte veiledning på lik linje med matte som hun forserte. Vi er strålende fornøyd med begge deler!"}},"chat":{"expressionError":"Noe gikk galt med dette utrykket","header":"Chat med privatlærer","infoBox":"Hallo! \u003cbr\u003e\u003cbr\u003e Still dine mattespørsmål her.","infoBoxEnd":"Tjenesten er kun gratis i en begrenset periode. Chatten besvares bare på engelsk.","infoBoxItem1":"1. Gjengi oppgaven.","infoBoxItem2":"2. Oppgi alle detaljer.","infoBoxItem3":"3. 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Her kan du komme for å se oppdrag og oppdateringer fra din mattereise. Der ute finner du ting som denne:","chestTitle":"Velkommen!","close":"Lukk","collection":"Min samling","collectionList":{"allItems":"Alle troféer","allRewards":"Alle belønninger","bootcamps":"Bootcamps","competition":"Konkurranse","mathdrill":"Drillhefter","mathEssentials":"Drillhefter"},"collectionShowcase":{"bootcamp":"Bootcamp","gathered":"Samlet","championship-2021":"Gangetabellkonkurransen 2021"},"rankPopupTitle":"Nytt XP-nivå-system!","rankPopupText":"Du tjener XP ved å gjøre aktiviteter i portalen.  Øk nivået ditt ved å nå neste XP-terskel.","weExchange":"Vi veksler","completed":"Fullført","dummyData":{"mult":{"title":"Ganging","description":"Hva er ganging"},"algebra":{"title":"Algebra","description":"Elementær algebra"}},"event":{"checkQuest":"Jeg har lest og forstått","INTRO":{"dialogue":[{"text":"Dette er din base. 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La meg vise deg litt rundt først.","introTitle":"Bra start!","kudos":"Kudos","level":"Level","levelUp":"Level up","navbarIcon":"Space","next":"Neste","shadow":"Skygge","playerName":"Spillernavn...","questCompleted":"Matteoppdrag fullført","questDescriptions":{"bootcamp":"Utforsk bootcampen \u003cb\u003e{{BootcampTitle}}\u003c/b\u003e og bekjemp bossen!","essential":"Fullfør drillheftet og bevis dine ferdigheter!","encyclopedia":"Les og forstå \u003cb\u003e{{EncyclopediaArticle}}\u003c/b\u003e og kryss av i boksen nederst!"},"quests":"Mine matteoppdrag","mathQuests":"Matteoppdrag","completedMathQuests":"Fullførte oppdrag","completeQuestsToSee":"Fullfør dine matteoppdrag for å se dem her","questText":"Ta quizen for å få en tilpasset læringssti","quiz":{"advancedQuizText":"For å få flere matteoppdrag kan du velge deg ut noen emner du vil lære mer om!","advancedQuizTitle":"Godt jobba!","advancedSubjects":{"quiz-subject-advanced-derivatives":{"title":"Derivasjon"},"quiz-subject-advanced-differential-equations":{"title":"Differensiallikninger"},"quiz-subject-advanced-integrals":{"title":"Integraler"},"quiz-subject-advanced-logarithms":{"title":"Logaritmer"},"quiz-subject-advanced-pi":{"title":"Pi, π"},"quiz-subject-advanced-probability":{"title":"Sannsynlighet"},"quiz-subject-advanced-proofs":{"title":"Bevis"},"quiz-subject-advanced-quadratic-equations":{"title":"Andregradslikninger"},"quiz-subject-advanced-series":{"title":"Rekker"},"quiz-subject-advanced-sets-of-numbers":{"title":"Tallmengder"},"quiz-subject-advanced-trigonometry":{"title":"Trigonometri"},"quiz-subject-advanced-vectors":{"title":"Vektorer"}},"applyText":"Du vet åpenbart mye om matte! Hvis du vil bli med på House of Math teamet, så kan du søke \u003clinkToCareers\u003eher\u003c/linkToCareers\u003e.","awardTitle":"Vi spillifiserer din opplevelse","awardTitleSubtext":"Underveis vil du finne kister som denne","awesome":"Kult!","buttonTakeQuiz":"Ta quizen","creatingYourLearningPath":"Genererer din optimale læringssti...","learningPath":"Genererer din optimale læringssti","greatResultsText":"Nå må du bare velge de emnene du allerede kjenner til for å fortsette","greatResultsTitle":"Flotte resultater på quizen!","subjectKnowledgeRating":{"0":"Dårlig","1":"Litt dårlig","2":"Nøytral","3":"Godt","4":"Veldig godt"},"subjects":{"quiz-subject-algebraic-factorization":{"questionRating":"Algebraisk faktorisering","title":"Algebraisk faktorisering"},"quiz-subject-basic-algebra":{"questionRating":"Hvor godt mestrer du bokstavregning?","title":"Bokstavregning "},"quiz-subject-equations":{"questionRating":"Hvor godt mestrer du likninger?","title":"Likninger"},"quiz-subject-fractions":{"questionRating":"Hvor godt mestrer du brøkregning?","title":"Brøk"},"quiz-subject-functions":{"questionRating":"Hvor godt mestrer du funksjoner?","title":"Funksjoner"},"quiz-subject-parentheses":{"questionRating":"Hvor godt mestrer du parenteser?","title":"Parenteser"},"quiz-subject-percent":{"questionRating":"Hvor godt mestrer du prosentregning?","title":"Prosent"},"quiz-subject-powers":{"questionRating":"Hvor godt mestrer du potensregning?","title":"Potenser"},"quiz-subject-the-multiplication-table":{"questionRating":"Hvor godt mestrer du gangetabellen 1-10?","title":"Gangetabellen"}},"title":"For effektiv læring, ta quizen! ","titleSubtext":"Ta denne quizen for å finne ut hvor du skal begynne.","yesNoQuestionTitle":"Har du jobbet med \u003clowercase\u003e{{ subject }}\u003c/lowercase\u003e?"},"retakeQuiz":"Ta quizen på nytt","reward":"Belønning","storyCards":{"buttonCool":"Kult!","buttonIWill":"Det skal jeg!","buttonJump":"Hopp","buttonLetUsTry":"La oss prøve!","buttonNice":"Herlig!","buttonOfCourse":"Selvfølgelig!","buttonOh":"Oi!","buttonPeekaBoo":"Titt tei","buttonRoar":"Brøl","buttonThanks":"Takk!","buttonWow":"Wow!","buttonYes":"Ja!","ChoosePetTrick":"Dragen din er klar for å lære seg et nytt triks! Trykk på en av knappene nedenfor!","ChooseYourEgg":"Se! Noen har lagt igjen en gave til deg!","CoolDown":"Du må vente litt før du kan gi mer Mattenergi! Bruk tiden til å samle inn mer Mattenergi fra kistene i Drillheftene!","CoolDownFirstTime":"Egget ditt er mett, og vil ikke ha mer Matteenergi nå! 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... Egget ditt virker som om det er sultent!","EggHungry2":"Her ta litt Kudos!","EggHungry3":"Her, ta litt Kudos! Prøv å gi Kudos til egget ved å klikke på det!","EggInfoStillHatching":"Egget ditt er under klekking! Men du må sørge for å få gitt det mer Kudos!","EggIsAboutToHatch":"Utrolig! Du har fått en drage i basen din!","FeedMathEnergyToEggFirstTime":"Prøv å gi Kudos til egget ved å klikke på det!","FirstTimeGettingMathEnergyFromChest":"Kult! Du har funnet din første kule med KudosEnergi! Gi det til egget du har i basen!","GiveDragonName":"Skriv inn navnet til dragen din eller trykk OK.","InfoAboutPetTrick":"Dragen din er villig til å lære når den er mett og fornøyd!","MathEnergyTutorial1":"Du må gi nok Kudos til egget for å få det til å klekke!","MathEnergyTutorial2":"Hvis du gjør matteoppdrag så kan du finne mer Kudos!","NextHatchingState1":"Oi, se! Egget ditt ser litt annerledes ut!","NextHatchingState2":"Egget ditt har sprukket enda litt mer! 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