Problem

Monday 26 April 2004 at 4:53 pm | In Articles | 4 Comments

Firstly, show that:

    \log_{a^n}b^n=\log_{a}b

Secondly, are there any other operations you can do to a and b and the \log retains the same value? In other words:

    Find all functions f,g:\mathbb{R^+}\mapsto \mathbb{R^+} such that \log_{f(a)}g(b)=\log_a b for all a,b\in\mathbb{R^+}

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  1. lhs = n \log_{a^n} b
    = n \over \log_b a^n
    =n / (n \times \log_b a )
    = \log_a b
    = rhs

    Comment by msk — Tuesday 11 May 2004 6:44 pm #

  2. Yes – that’s right. Another way to do it is to use the change of base formula to change the base of the LHS to a.

    Can you solve the second problem?

    Comment by steve — Tuesday 11 May 2004 6:58 pm #

  3. Require that
    (f(x))^n = f(x^n)
    ?

    Comment by Ronald — Friday 11 March 2005 2:34 pm #

  4. In fact f(x)=g(x)=x^k for some k \in \mathbb{R}

    Comment by Steve — Friday 11 March 2005 3:08 pm #

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