Question

A student solved the equation below by graphing.

log Subscript 6 Baseline (x minus 1) = log Subscript 2 Baseline (2 x + 2)

Which statement about the graph is true?
The curves do not intersect.
The curves intersect at one point.
The curves intersect at two points.
The curves appear to coincide.

Answer 1

The curves do not intersect.

Reason: Let h(x) = log_6(x−1) − log_2(2x+2), defined for x>1. Then
h'(x) = 1/((x−1) ln6) − 1/((x+1) ln2)
= [(1−x) ln3 + 2 ln2] / [(x−1)(x+1) ln6 ln2].

h'(x) = 0 at x = 1 + 2 ln2 / ln3 ≈ 2.26186, where h has its single maximum. Evaluating there gives
h(1 + 2 ln2 / ln3) ≈ 0.12945 − 2.70510 ≈ −2.5756 < 0.

Since h(x) → −∞ as x → 1+ and as x → ∞, and its maximum is negative, h(x) is always negative on its domain, so log_6(x−1) = log_2(2x+2) has no solution.