# How do you solve log x^3=(log x)^3 ?

## I know there are 3 solutions, but I don't know the steps taken to solve it.

Jul 17, 2017

$\log {x}^{3} = {\left(\log x\right)}^{3}$

$\implies \log {x}^{3} = {\left(\log x\right)}^{3}$

$\implies 3 \log x = {\left(\log x\right)}^{3}$

let $\log x = y$

then the equation becomes

${y}^{3} - 3 y = 0$

$\implies y \left({y}^{2} - 3\right) = 0$

When $y = 0$

$\implies \log x = 0 = \log 1$

$\implies x = 1$

when

${y}^{2} - 3 = 0$

$\implies y = \pm \sqrt{3}$

$\implies \log x = \pm \sqrt{3}$

$\implies x = {e}^{\pm \sqrt{3}}$