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Math Help - Logs

  1. #1
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    Logs

    Given that logab(a)=4, calculate logab 3√a /√b. Start with the exponent property of logs, then the quotient rule, and then change of base.
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  2. #2
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    Hello KingV15
    Quote Originally Posted by KingV15 View Post
    Given that logab(a)=4, calculate logab 3√a /√b. Start with the exponent property of logs, then the quotient rule, and then change of base.
    I'm guessing that the question is:
    Given \log_{ab}(a) = 4, calculate \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right)
    (The alternative is that you have to calculate \log_{ab}\left(\frac{3\sqrt{a}}{\sqrt{b}}\right), which will leave a term in \log_{ab}(3) at the end.)

    So, if my assumption is correct:
    \log_{ab}(a) = 4

    \Rightarrow (ab)^4 = a

    \Rightarrow a^4b^4 = a

    \Rightarrow b^4 = a^{-3}

    \Rightarrow b = a ^{-\frac34}
    And then:
    \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right)=\tfrac13\log_{ab}(a) - \tfrac12\log_{ab}(b)
    =\tfrac13\log_{ab}(a) - \tfrac12\log_{ab}(a^{-\frac34})

    =\tfrac13\log_{ab}(a) - \tfrac12\times(-\tfrac34)\log_{ab}(a)

    =(\tfrac13+\tfrac38)\log_{ab}(a)

    =\frac{17}{24}\times 4

    =\frac{17}{6}
    (The answer to the alternative question is, using a similar method, \frac72+\log_{ab}(3).)
    Grandad
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