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Math Help - Messing with the bases of logarithms

  1. #1
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    Exclamation Messing with the bases of logarithms

    if log base a of x =c , and log base b of x = d, then what is the general statement that expresses log base (ab) of x in terms of c and d? plz help.
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  2. #2
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    If the problem state something like
    \log_a x = c
    \log_b x = d
    \log_{ab} x = ?

    If so, you will want to use a change of base rule here, i.e.
    \frac{\log_{a} x}{\log_{ab} a} = \log_{ab} x
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by 3k1yp2 View Post
    if log base a of x =c , and log base b of x = d, then what is the general statement that expresses log base (ab) of x in terms of c and d? plz help.
    Use change of base formula:

    \log_a x=c\implies c=\frac{\ln x}{\ln a}\implies \ln a=\frac{\ln x}{c} and \log_b x=d\implies d=\frac{\ln x}{\ln b}\implies \ln b=\frac{\ln x}{d}.

    Now, \log_{ab}x=\frac{\ln x}{\ln\!\left(ab\right)}=\frac{\ln x}{\ln a+\ln b}=\frac{\ln x}{\frac{1}{c}\ln x+\frac{1}{d}\ln x}=\frac{1}{\frac{c+d}{cd}}=\frac{cd}{c+d}

    Does this make sense?
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  4. #4
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    i do need to know what log base ab of x equals, but it has to be in terms of c and d.
    i got that
    log base a of x = c
    log base b of x = d
    so a^c = b^d , but im stuck there.
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  5. #5
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    that was extremely helpful, but i did get lost at the part where
    ln(x) was by the sum of (1/c ln(x) ) and (1/d ln(x) ). where or how did you get those expressions in the denominator?


    Nevermind, now i get it!
    thank youuu! >singing<
    Last edited by 3k1yp2; November 19th 2009 at 06:35 PM. Reason: to Chris L T521
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