MathDB
Solving for the Log

Source: 2017 AMC 12A #20

February 8, 2017
AMCAMC 12AMC 12 A2017 AMC 12A

Problem Statement

How many ordered pairs (a,b)(a, b) such that aa is a real positive number and bb is an integer between 22 and 200200, inclusive, satisfy the equation (logba)2017=logb(a2017)(\log_b a)^{2017} = \log_b (a^{2017})?
<spanclass=latexbold>(A) </span>198<spanclass=latexbold>(B) </span>199<spanclass=latexbold>(C) </span>398<spanclass=latexbold>(D) </span>399<spanclass=latexbold>(E) </span>597 <span class='latex-bold'>(A) \ </span>198\qquad <span class='latex-bold'>(B) \ </span> 199 \qquad <span class='latex-bold'>(C) \ </span> 398 \qquad <span class='latex-bold'>(D) \ </span>399\qquad <span class='latex-bold'>(E) \ </span> 597