Solve for x in the following logarithm problem.

Question

log_4 (x^2+1)= ½ - log_4 (2-x)

Show all work.

Answer 1

log₄(x²+1) = 1/2-log₄(2-x)

log₄(x²+1) + log₄(2-x) = 1/2

log₄((x²+1)(2-x)) = 1/2

(x²+1)(2-x) = -x³ +2x²-x +2 = 4^1/2 = ±2

Case 1: Use +2

-x³ +2x²-x +2 = 2

x³-2x²+x = x(x²-2x+1) = x(x-1)(x-1) = 0 --> x = 0, 1, 1

Case 2: Use -2

-x³ +2x²-x +2 = -2

-x³ +2x²-x + 4 = 0

x³ -2x² + x - 4 = 0

This can't be factored!