TITLE: A question about simplification of an expression involving log: $16777216^\frac{\log(64n)}{3\log(4)}$
QUESTION [3 upvotes]: I have a practice question regarding simplification of the following expression:
$16777216^\frac{\log(64n)}{3\log(4)}$
So I have tried to do this:
$(64^4)^\frac{\log64n}{\log64}$
and now I got stuck. Maybe there are something more I could do to the exponent term using change of base.
Say I simplify it further to:
$(64^4)^{\log_{64}{(64n)}}$
But this is as far as I can go, I think there might be more simplification that could be done. But not sure how to proceed further. Could someone help a bit?

REPLY [2 votes]: Just switch the exponents around as
$$(a^b)^c = a^{bc} = (a^c)^b \implies \left( 64^4 \right)^{\log_{64}(64n)} = \left( 64^{\log_{64}(64n)} \right)^4=(64n)^4 = 16,777,216 \cdot n^4$$

REPLY [0 votes]: $(64^4)^{\log_{64}64n} = (64^4)^{1+\log_{64}n} = 64^4 \times (64^4)^{\log_{64}n} = 64^4 \times (64^{\log_{64}n})^4 = 64^4n^4$
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