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Math Help - Logs

  1. #1
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    Logs

    If logn(108)=a, and logn(72)=b, find logn(2)in terms of a and b.
    Consider logn(32).
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  2. #2
    MHF Contributor
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    Hello Tiger
    Quote Originally Posted by Tiger View Post
    If logn(108)=a, and logn(72)=b, find logn(2)in terms of a and b.
    Consider logn(32).
    \log_n(108)=a
    \Rightarrow n^a=108=2^2\times3^3 ...(1)
    \log_n(72) = b
    \Rightarrow n^b=72=2^3\times3^2 ...(2)
    x=\log_n(2)
    \Rightarrow n^x= 2 ...(3)
    So we need to eliminate the 3's from equations (1) and (2), to be left with a power of 2 only. So square both sides of (1) and cube both sides of (2):
    n^{2a} = 2^4\times 3^6
    and
    n^{3b} = 2^9\times 3^6
    Divide:
    \frac{n^{3b}}{n^{2a}}= \frac{2^9\times 3^6}{2^4\times 3^6}

    \Rightarrow n^{3b-2a} = 2^5=\Big(n^x\Big)^5, from
    (3)

    \Rightarrow x = \tfrac15(3b-2a)
    Grandad
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