Example 9. Solve the equation

$$
5^{2 x}-2 \cdot 5^{x}-15=0
$$
Solution. Let $t=5^{x}$. Then $t^{2}-2 t-15=0$. From this, we find $t_{1}=5, t_{2}=-3$. Thus, the given equation is equivalent to the system of equations

$$
5^{x}=5, \quad 5^{x}=-3
$$

## § 3. EXPONENTIAL EQUATIONS

The second equation of this system has no roots, since $-3<0$ for any $x$, while from the first equation we find that $x=1$ is the only root of the original equation.

An equation of the form

$$
a^{f(x)}=b
$$

where $a>0, a \neq 1, b>0$, can be solved by taking the logarithm of both sides (this is possible since both sides of the equation are positive). Taking the logarithm, we obtain the equation

$$
f(x)=\log _{a} b,
$$

which is equivalent to equation (3).