Example 21. Solve the equation

$$
1+3^{x / 2}=2^{x}
$$
Solution. Dividing both sides of the equation by $2^{x}$, we obtain the equation

$$
(1 / 2)^{x}+(\sqrt{\overline{3}} / 2)^{x}=1
$$

equivalent to equation (21). Equation (22) can be written as

$$
\left(\sin \frac{\pi}{6}\right)^{x}+\left(\cos \frac{\pi}{6}\right)^{x}=1
$$

Comparing this equation with the fundamental trigonometric identity, we conclude that the number $x_{1}=2$ is a root of equation (21). There are no other roots, since in the left part of equation (22) there is a sum of two decreasing exponential functions $y=(1 / 2)^{x}$ and $y=(\sqrt{3} / 2)^{x}$.

Some exponential equations contain expressions of the form

$$
f(x)^{g(x)}
$$

By definition, we consider that

$$
f(x)^{g(x)}=10^{g}(x) \lg f(x)
$$

therefore, the function $f(x)^{g(x)}$ makes sense only when both functions $f(x)$ and $g(x)$ are defined and $f(x)>0$.