The number of real solutions of the equation $\sin \left( e ^ { x } \right) = 5 ^ { x } + 5 ^ { - x }$ is ____________________.

$\text { Let } \begin{aligned} y_{1} & =\sin \left(\mathrm{e}^{\mathrm{x}}\right) \\ \mathrm{y}_{2} & =5^{\mathrm{x}}+5^{-\mathrm{x}} \end{aligned}$

Now, $0<\mathrm{e}^{\mathrm{x}}<\infty$

$\therefore-1 \leq \sin \left(\mathrm{e}^{\mathrm{x}}\right) \leq 1$

$\therefore\left(\mathrm{y}_{1}\right)_{\max }=1$

$\therefore\left(\mathrm{y}_{1}\right)_{\text {min }}=-1$

$\mathrm{y}_{2}=5^{\mathrm{x}}+5^{-\mathrm{x}}=5^{\mathrm{x}}+\frac{1}{5^{\mathrm{x}}}$

Now, $5^{\mathrm{x}}>0$

Let $5^{\mathrm{x}}=\mathrm{t}$

$\therefore \mathrm{y}_{2}=\mathrm{t}+\frac{1}{\mathrm{t}}$

Now, $t+\frac{1}{t} \geq 2$ if $\quad t>0$

Hence, $\mathrm{y}_{2} \geq 2$

$\Rightarrow\left(\mathrm{y}_{2}\right)_{\min }=2$

Therefore, $\mathrm{y}_{1} \neq \mathrm{y}_{2}$

$\therefore$ No real solutions of the equation.

$\sin \left(e^{x}\right)=5^{x}+5^{-x}$ has zero solution.