How are you able to simplify 3^(2log3^5)?

Question

How are you able to simplify 3^(2log3^5)?

Answer 1

I'm going to take a wild guess that you mean the exponent to be 2 times log base 3 of 5, and not 2 times the common log of 3 to the fifth.

Remember your rules of logs:

a log b = log (b^a)

I'm just going to use "log" for "log base 3" right now.

So, 2 log 5 = log 5^2 = log 25

Now:  you have 3 to the power log base 3 of 25.  Remember another rule of logs:

10^log x = x

assuming the log is log base 10.

So, 3^log(base 3)25 = 25.