The number of solutions of the equation _4(x-1)=_2(x-3) is Answer: 1

Question

The number of solutions of the equation $\log_4(x-1)=\log_2(x-3)$ is

Answer: 1

Accepted Answer

$$\log_4(x-1)=\log_2(x-3) \implies \tfrac12\log_2(x-1)=\log_2(x-3) \implies \log_2(x-1)=\log_2(x-3)^2.$$

So $x-1=(x-3)^2 \implies x^2-7x+10=0 \implies (x-5)(x-2)=0$, giving $x=2,5$. Since $x>3$ is required (domain), $x=2$ is rejected. Hence $x=5$ is the only solution --- 1 solution.

Solutions · 1