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Fahrenheit 45one solution to logarithm equation

Source: 2017 AIME II #7

March 23, 2017
logarithmsAMCAIMEAIME IIgenius title

Problem Statement

Find the number of integer values of kk in the closed interval [500,500][-500,500] for which the equation log(kx)=2log(x+2)\log(kx)=2\log(x+2) has exactly one real solution.