Question

Solve log81 32x - log81(x-3)=3/4 for x

Answer 1

1 step. Note that

$x>0,\\ \\x-3>0\Rightarrow x>3.$

Therefore, possible x are $x>3.$

2 step. Use property $\log_ab-\log_ac=\log_a\dfrac{b}{c}.$

Then

$\log_{81}32x-\log_{81}(x-3)=\log_{81}\dfrac{32x}{x-3}.$

3 step.

$\log_{81}\dfrac{32x}{x-3}=\dfrac{3}{4}\Rightarrow 81^{\log_{81}\frac{32x}{x-3}}=81^{\frac{3}{4}},\\ \\\dfrac{32x}{x-3}=(3^4)^{\frac{3}{4}},\\ \\\dfrac{32x}{x-3}=27,\\ \\32x=27(x-3),\\ \\32x=27x-81,\\ \\5x=-81,\\ \\x=-\dfrac{81}{5}.$

4 step. Since $-\dfrac{81}{5}$ this solution is extra.

Answer: no solution, choice D.