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How do you simplify the following expression? $$\log\left(3^{(5^7)}\right)$$ I know that logarithms are like the inverse of exponents, but are there any tricks to simplify powers inside logarithms?

Edit: There are five answer choices:

(A) $7\log(3^5)$

(B) $35\log(3)$

(C) $7(\log(3))(\log(5))$

(D) $7(\log(3)+\log(5))$

(E) $5^7\log(3)$

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  • $\begingroup$ Fun how choice A and B are identical. $\endgroup$ Commented May 18, 2014 at 6:16

2 Answers 2

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As $\displaystyle \log(a^m)=m\log a$ where both $\log$ remain defined

Here $a=3, m=5^7$

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$\log 3^{(5^7)}=5^7\log 3$

The power on the inside become multiplication on the outside, and becomes the simplest form (unless you want to turn $5^7$ into a single long number instead of that power)

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