7.186. $\log _{12}\left(4^{3 x}+3 x-9\right)=3 x-x \log _{12} 27$.
## Solution.

Domain of definition: $4^{3 x}+3 x-9>0$.

Rewrite the equation as

$$
\log _{12}\left(4^{3 x}+3 x-9\right)+\log _{12} 27^{x}=3 x \Rightarrow \log _{12} 27^{x}\left(4^{3 x}+3 x-9\right)=3 x
$$

from which $27^{x}\left(4^{3 x}+3 x-9\right)=12^{3 x} \Leftrightarrow 4^{3 x}+3 x-9=4^{3 x}, 3 x-9=0, x=3$.

Answer: 3.