Random UTube Thumbnail Question

 Scrolling through my UTube feed I spot a thumbnail with a problem statement:

Find all the real solutions to:

$$3^x . 8^{\frac{x}{x+2}}=6$$

Initial observations: $x=1$ is a solution, and $x=-2$ is not a solution.

An obvious thing to do with this is to take logs:

$$x\log(3)+\frac{x}{x+2}\log(8)=\log(6)$$

We can multiply through by $(x+2)$ to get a quadratic in $x$

$$x^2 \log(3) +x (\log(3)+2\log(2))-2(\log(3)+\log(2))=0$$

or:

$$x^2+x\left(1+2\frac{\log(2)}{\log(3)}\right) -2\left(1+\frac{\log(2)}{\log(3)}\right)=0$$

As this is a quadratic with one know real root, the other root is also real, and so there are two real solutions (assuming distinct roots).

But we know that $x=1$ is a root and so $(x-1)$ is a factor, so if $x=u$ is the other root we must have 

$$u=-2\left(1+\frac{\log(2)}{\log(3)}\right) \approx -3.262$$

Now we check that this is a solution (in this case using Gnu-Octave, but a calculator will do...)

>> x=-2*(1+log(2)/log(3))
x = -3.2619
>
>> 3^x*8^(x/(x+2))
ans = 6.0000