2015 AMC 12B Problem 8

Problem 8 of 25EasierAlgebra

What is the value of

\left(625^{\log_5 2015}\right)^{\frac14}?

Show solution

Solution

Since

625 = 5^4,

we have

\begin{gathered} 625^{\log_5 2015} = 5^{4\log_5 2015} \\ = \left(5^{\log_5 2015}\right)^4 = 2015^4. \end{gathered}

Taking the fourth root gives

\left(2015^4\right)^{\frac{1}{4}} = 2015.

Thus, the correct answer is D.

Concepts: logarithm · exponent

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.