Solving logarithmic equations The equation that I'm trying to solve is:
$$\log _{5x+9}(x^2+6x+9)+\log _{x+3}(5x^2+24x+27)=4$$
Using algebra and principles of logarithms I managed to get the equation down to $$\frac{2\left(\log \left(x+3\right)\right)^2+\left(\log \left(5x+9\right)\right)^2}{\log \left(x+3\right)\log \left(5x+9\right)}=3$$
 assuming that everything I've done so far is correct. I'm not sure what to do from here. I think there might be some kind of perfect square to factor but I'm not sure how I would even factor it from this situation (or if it would help).
Any suggestions?

For the sake of simplification, set $a=\log (x+3)$ and $b=\log (5x+9)$ then the equation you obtained writes as $2a^2-3ab+b^2=0$. Using standard factoring this is $(a-b)(2a-b)=0$. I think you can do the rest!
