Example 20. Solve the equation

$$
3 x^{\log _{5} 2}+2^{\log _{5} x}=64
$$
Solution. The domain of the equation: $x>0$. On this set

$$
x^{\log _{5} 2}=2^{\log _{5} x}
$$

therefore, the given equation is equivalent to the equation

$$
3 \cdot 2^{\log _{5} x}+2^{\log _{5} x}=64
$$

i.e., the equation

$$
2^{\log _{5} x}=16
$$

From this, we get $\log _{5} x=4$, i.e., $x=625$. The number 625 belongs to the domain of the original equation and, therefore, is its only root.

Sometimes it is advisable to take the logarithm of both sides of the equation to reduce its solution to one of the simplest equations. As a rule, this has to be done for mixed-type equations containing both exponential and logarithmic functions.