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(b)/(c).`

Then

`\log_(81)32x-\log_(81)(x-3)=\log_(81)(32x)/(x-3).`

3 step.

`\log_(81)(32x)/(x-3)=(3)/(4)\Rightarrow 81^{\log_(81)(32x)/(x-3)}=81^{(3)/(4)},\\ \\(32x)/(x-3)=(3^4)^{(3)/(4)},\\ \\(32x)/(x-3)=27,\\ \\32x=27(x-3),\\ \\32x=27x-81,\\ \\5x=-81,\\ \\x=-(81)/(5).`

4 step. Since
`-(81)/(5)<3,` this solution is extra.

Answer: no solution, choice D.