Example 13. Solve the equation

$$
3 \cdot 16^{x}+37 \cdot 36^{x}=26 \cdot 81^{x}
$$
Solution. In this equation, the numbers $16,36,81$ form three consecutive terms of a geometric progression (with a common ratio of $9 / 4$).

To solve the original equation, divide both sides by $81^{x}$. We get

$$
3 \cdot(4 / 9)^{2 x}+37 \cdot(4 / 9)^{x}-26=0
$$

Let $t=(4 / 9)^{x}$; then equation (8) becomes

$$
3 t^{2}+37 t-26=0
$$

from which $t_{1}=2 / 3$ and $t_{2}=-13$.

Thus, equation (7) is equivalent to the system of two exponential equations:

$$
(4 / 9)^{x}=2 / 3, \quad(4 / 9)^{x}=-13
$$

The second equation of this system has no solutions,

From the equation $(4 / 9)^{x}=2 / 3$, i.e., the equation

$$
(2 / 3)^{2 \boldsymbol{x}}=2 / 3
$$

we find the only root of the original equation: $x_{1}=1 / 2$.