Exponential equation - logarithmisation is the transformation of this equation: $$9^x + 6^x = 2× 4^x$$
into this: $$\log_2 (9^x) + \log_2 (6^x)=\log_2 (2×4^x)$$ correct? I want to know because I really want to solve this equation.

It's $f(x)=0$, where $$f(x)=\left(\frac{3}{2}\right)^{2x}+\left(\frac{3}{2}\right)^{x}-2.$$
We see that $f$ increases, which says that our equation has one root maximum.
But, $0$ is a root and we are done!
Your reasoning is wrong because $\log(a+b)$ is not always equal to $\log{a}+\log{b}.$
