Example 1. Solve the equation

$$
4^{x}=8^{2 x-3}
$$
Solution. Since

$$
4^{x}=\left(2^{2}\right)^{x}=2^{2 x}, \quad 8^{2 x-3}=\left(2^{3}\right)^{2 x-3}=2^{6 x-9}
$$

then

$$
2^{2 x}=2^{6 x-9} \Leftrightarrow 2 x=6 x-9 \Leftrightarrow x=9 / 4
$$