Answer
Solution set = $\{0.01,10^{3/2}\}$
Work Step by Step
The equation is defined only for $ x\gt 0$.
Temporarily, substitute $ t=\log x $.
We have
$ t(2t+1)=6$
$2t^{2}+t-6=0$
Factor by finding two factors of $2(-6)=-12$ with a sum of 1. We find -3 and +4. Rewrite and factor in pairs.
$2t^{2}+4t-3t-6=0$
$2t(t+2)-3(t+2)=0$
$(t+2)(2t-3)=0$
$\left[\begin{array}{lll}
t=-2\Rightarrow & \log x=-2\Rightarrow & x=10^{-2}=0.01\\
or & & \\
t=3/2\Rightarrow & \log x=3/2\Rightarrow & x=10^{3/2}\approx 31.62
\end{array}\right]$
Solution set = $\{0.01,10^{3/2}\}$
Check graphically:
