Prove equality $a^{\log_b c} = c^{\log_b a}$

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Taking the logarithm to base $b$ of both sides, the equation is equivalent to $$(\log_bc)(\log_ba)=(\log_ba)(\log_bc)\ .$$

I would say $n^{\log_23}$ is better because it makes it clear that it is $n$ to a constant power. This is of course also true for $3^{\log_2n}$, but it's not so obvious.

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Updated on March 11, 2020

Comments

  • Solsma Dev
    Solsma Dev over 3 years

    I'm try to prove the equality:

    $$a^{\log_b c} = c^{\log_b a}$$

    I'm having trouble finding information regarding this, also I need to figure out why $n^{\log_2 3}$ is better than $3^{\log_2 n}$ as a closed form formula for $M(n)$? Thank you!