Example 8. Solve the equation

$$
4^{x}-3^{x-1 / 2}=3^{x+1 / 2}-2^{2 x-1}
$$
Solution. Group the terms containing powers with base 4 and base 3:

$$
4^{x}+(1 / 2) 4^{x}=3^{x+1 / 2}+3^{x-1 / 2}
$$

Factor out the common factors:

$$
4^{x}(1+1 / 2)=3^{x-1 / 2}(1+3)
$$

Divide this equation by the expression on its right side, we get $(4 / 3)^{x-3 / 2}=1$. From this, we find $x-3 / 2=0$; hence, $x=3 / 2$ is the only root of the original equation.

An equation of the form

$$
f\left(a^{x}\right)=0
$$

can be reduced to solving an equivalent system of simple exponential equations by the substitution $t=a^{x}$:

$$
a^{x}=t_{1}, \quad a^{x}=t_{2}, \ldots, a^{x}=t_{k}
$$

where $t_{1}, t_{2}, \ldots, t_{k}$ are all the roots of the equation $f(t)=0$. For example, the equation

$$
A a^{2 x}+B a^{x}+C=0
$$

where $A, B, C$ are some numbers, $a>0, a \neq 1$, can be reduced to solving the equivalent system of equations $a^{x}=t_{1}, a^{x}=t_{2}$, where $t_{1}$, $t_{2}$ are the roots of the equation $A t^{2}+B t+C=0$.