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Math Help - Different base logarithm proof

  1. #1
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    Different base logarithm proof

    Hello!

    I need to prove that log_a(x) = c * log_b(x)

    I don't seem to know how to start proving this very evident example.

    I started to move them around, and change bases, but got nothing useful.
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  2. #2
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    stgo
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    Quote Originally Posted by sioux View Post
    Hello!

    I need to prove that log_a(x) = c * log_b(x)

    I don't seem to know how to start proving this very evident example.

    I started to move them around, and change bases, but got nothing useful.

    \log_a(x)=\dfrac{\log(x)}{\log(a)}=\underbrace{\df  rac{\log(b)}{\log(a)}}_{c}\cdot \dfrac{\log(x)}{\log(b)}=c\cdot \log_b(x)
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  3. #3
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    Haha, was right in front of me. Thanks!
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  4. #4
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    If you did not know that formula, you could use the fact that log_a(x) is defined as the inverse function to a^x. That is, that log_a(a^x)= x and a^{log_a(x)}= x.

    In particular, if y= log_b(x) then x= b^y. But then x= a^{log_a(b^y)}= a^{ylog_a()}. From that, log_a(x)= log_a(a^{ylog_a(b)}= (log_a(b))y= (log_a(b))log_b(x). That is log_a(x)= c log_b(x) with c= log_a(b).
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