>Logaritmelikninger>Hvordan løse eksponentiallikninger

Hvordan løse eksponentiallikninger

En eksponentiallikning er en likning der $x$ er en eksponent. For å løse disse må du bruke logaritmer for å få $x$ ned fra eksponenten. Du velger selv om du vil bruke $\log$ eller $\ln$ . For å jobbe med eksponential- og logaritmelikningene er det viktig å huske logaritmereglene og potensreglene! Du løser disse likningene på følgende måter:

Regel

Eksponentiallikninger

De samme reglene gjelder uavhengig om du bruker

\ln

eller

\lg

. Du vil nok for det meste bruke

\ln

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

Ta logaritmen på begge sider for å få $x$ ned:

x \ln a = \ln b

Del på $\ln a$ på begge sider av likheten:

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

NB! Svaret blir det samme uavhengig om du bruker $\ln$ eller $\lg$ !

Eksempel 1

Løs likningen $1 0^{x} = 3000$

\begin{array}{l} 1 0^{x} = 3000 \ln 1 0^{x} & = \ln 3000 x & = \frac{\ln 3000}{\ln 10} x & \approx 3, 477Eksempel 2 Løs likningen 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 \ln 2}{\ln 2} = 2Eksempel 3 Løs likningen 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}Vil du vite mer? Registrer deg Det er gratis!Neste oppslag Hvordan løse logaritmelikningerTilbakeHjem 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 disse må du bruke logaritmer for å få \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/math\u003e ned fra eksponenten. Du velger selv om du vil bruke \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e \u003cmi class=\"loglike\"\u003elog\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003c/math\u003e eller \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     For å jobbe med eksponential- og logaritmelikningene er det viktig å huske \u003ca href=\"/no/encyclopedia/tall-og-maaling/regning/potenser-og-rotter/potensregler\"\u003elogaritmereglene\u003c/a\u003e og \u003ca href=\"/no/encyclopedia/tall-og-maaling/regning/potenser-og-rotter/potensregler\"\u003epotensreglene\u003c/a\u003e! Du løser disse likningene på følgende måter: \u003c/p\u003e      \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-633\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Regel\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\"\u003eEksponentiallikninger\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eDe samme reglene gjelder uavhengig om 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 eller \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003c/math\u003e. 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                          \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-635\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e \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 likningen \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\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\" stretchy=\"false\"\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\" mathvariant=\"bold-italic\"\u003e\u003cmtable  displaystyle=\"true\" columnalign=\"left\" class=\"align-star\"\u003e                      \u003cmtr\u003e\u003cmtd  columnalign=\"right\" 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class=\"align-odd\"\u003e\u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\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\" stretchy=\"false\"\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-punc\" stretchy=\"false\" type=\"divide\"\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                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Abonner nå for å få tilgang på våre bootcamper og forbedre dine matteferdigheter."},"header":{"title":"Lær matte smartere og enklere","joinToGetAccess":"Bli med for å få tilgang","login":"Logg inn","buyNow":"Kjøp nå","joinNow":"Bli med nå","homTeam":"House of Math Team","freeTrial":"Gratis prøveperiode","basic":"Basic","priceStartsFrom":"Priser fra","getBetterGrade":"Få bedre karakter i matte","totalMathSolution":"Møt din alt-i-ett matteløsning","trustedByTwoMillion":"Over 2 millioner \u003cbr\u003e fornøyde brukere","faq":"Ofte stilte spørsmål","subscriptionSection":"Få en bedre mattekarakter"},"pricing":{"productTitleYearly":"År","perMonth":"pr mnd","regularBilling":"Trekkes på en regelmessig basis","title":"Få en bedre mattekarakter","subTitle":"Fra 6 til 100+ år","month":"måned","year":"år","free":"Gratis","per":"pr","signUpForFree":"Registrer deg gratis","continueForFree":"Fortsett gratis","joinNow":"Bli med nå","startNow":"Start nå","changePlan":"Endre pakke","currentPlan":"Nåværende pakke","upgrade":"Oppgrader","bestValue":"Best pris","getAccessTo":"Få tilgang på","signUp":"Registrer deg","cancelAnytime":"Avslutt når som helst","couponPeriodPlural":"{{ priceLabel }} i {{ durationInMonths }} måned(er)","couponPeriodSingular":"{{ priceLabel }} i 1 måned","couponOnce":"{{priceLabel}} første måned, deretter","afterTime":"Så","freeTrialPeriodSingularCheckout":"{{ freeTrialPeriod }} dag gratis","freeTrialPeriodPluralCheckout":"{{ freeTrialPeriod }} dager gratis","freeTrialPeriodSingular":"1 dag gratis","freeTrialPeriodPlural":"{{ freeTrialPeriod }} dager gratis","seeMore":"Se mer","close":"Lukk","showAllPackages":"Vis alle pakker","showLessPackages":"Vis færre pakker","textYearly":"{{ price }} belastes årlig","testFreeTrialPeriod":"Prøv {{ trialDays }} dager gratis","startFreeTrialPeriod":"Start din {{ trialDays }}-dagers prøveperiode","unlockTrialPeriod":"Lås opp din {{ trialDays }}-dagers prøveperiode","then":"deretter","buy":"Kjøp","whatsIncluded":"Se hva som er inkludert","showLess":"Vis mindre","familyAccountComingSoon":"Familiekonto kommer snart"},"product":{"title":"Få din egen personlig læringssti","faq":"Ofte stilte spørsmål","faqLogo":"Ofte stilte spørsmål logo","teacher":"Jeg er en lærer","products":{"videoCourses":{"title":"Din sti for å få matten på gli","tagline":"Bootcamps","text":"Digitale kurs med instruerende videoer og interaktive oppgaver"},"keepTrack":{"title":"Få oversikt","tagline":"Statistikkpanel","text":"Ditt eget statistikkpanel for å følge med på din progresjon"},"unlimitedQuestions":{"title":"Still så mange spørsmål du vil","tagline":"24/7 mattechat","text":"Spør våre lærere så mange spørsmål du vil"},"excercises":{"title":"Alt du trenger på et sted","tagline":"Oppgaver","text":"Få tilgang på alt mattepensum fra barneskolen til videregående"}}},"checkout":{"title":"Subscription checkout","metaDescription":"Checkout"},"success":{"confirmation":"Bekreftelse","title":"Success","metaDescription":"Success","date":"Bestillingsdato","confirmationNumber":"Bekreftelsesnummer","welcome":"Velkommen til","fullAccess":"Du har nå tilgang til House of Math!","goHomeButton":"Gå til Hjem","backToUrl":"Tilbake til der du var","yourSubscriptionPackage":"Din abonnementspakke","trialPeriodEndsOn":"Prøveperioden slutter den","packageUpgraded":"Pakke oppgradert","packageChanged":"Pakke endret","checkmark":"Checkmark logo","getStarted":"Kom i gang"},"callback":{"text":"Du er nesten ferdig - vi behandler abonnementsplanen din"},"paymentFailed":{"message":"Vi kunne ikke behandle din betaling.\nPrøv igjen.","tryAgain":"Prøv igjen","refused":"Betalingen ble avvist","cancelled":"Betalingen ble kansellert","error":"Betalingen mislyktes","metaDescription":"Betalingen mislyktes","refusedSubtitle":"Vennligst prøv igjen med en annen betalingsmetode eller kort.","errorSubtitle":"Noe gikk galt under betalingingen.\nPrøv igjen senere."},"paymentPending":{"title":"Betalingen behandles","metaDescription":"Betalingen behandles","message":"Betalingen din er til behandling","messageEmailSent":"En bekreftelses epost vill bli sendt til \u003cemail\u003e{{email}}\u003c/email\u003e"},"modal":{"title":"Start gratis prøveperiode","text":"Takk for at du kommer tilbake til oss","trialUsed":"Start ditt abonnement nå!","button":"Få ubegrenset tilgang","freeVersion":"Fortsett å bruke gratisversjonen"},"onboarding":{"learningPath":"Genererer din optimale læringssti","choosePlan":"Velg en abonnementsplan som passer deg"},"testimonial":{"explore":"Få en dypere forståelse av \u003cbr\u003e alle mattekonseptene","stepByStep":"Trinnvise forklaringer som \u003cbr\u003e gjør det enkelt å forstå","provenMethod":"Bli bedre i matte med vår \u003cbr\u003e dokumenterte metode","reviewSilje":"Jeg er veldig fornøyd med oppfølgingen jeg har fått. 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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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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