Answer to Question #43114 in Algebra for Alice

Question

What value will make the equation true? Logx 81= -4
A) x= 1/9

B) x= 1/3

C) x=3

D) x=9

Answer 1

By definition of logarithm, the base of logarithm is a positive real number not equal to 1. The equality log_x81=-4 is equivalent to 81=x^(-4), that is 1/x⁴=81, or x⁴=1/81, x⁴-1/81=0. Seek a positive x, factor the left-hand side (x²-1/9)(x²+1/9)=0,
(x-1/3)(x+1/3)(x²+1/9)=0, x>0. The answer is x=1/3.