2015 AMC 12B Problem 8

Below is the professionally curated solution for Problem 8 of the 2015 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12B solutions, or check the answer key.

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Concepts:logarithmexponent

Difficulty rating: 1460

8.

What is the value of (625log52015)14?\left(625^{\log_5 2015}\right)^{\frac14}?

55

20154\sqrt[4]{2015}

625625

20152015

520154\sqrt[4]{5^{2015}}

Solution:

Since 625=54,625 = 5^4, we have 625log52015=54log52015=(5log52015)4=20154.625^{\log_5 2015} = 5^{4\log_5 2015} = \left(5^{\log_5 2015}\right)^4 = 2015^4.

Taking the fourth root gives (20154)1/4=2015.\left(2015^4\right)^{1/4} = 2015.

Thus, the correct answer is D.

Problem 8 in Other Years