## Task 17/79

Determine without numerical calculation which of the two numbers $e^{\pi}$ and $\pi^{e}$ is the larger one!
It is well known that the function $y=f_{1}(x)=x$ grows faster than the function $y=f_{2}(x)=\ln x$ for $x>1$. Proof of this claim: It is $f_{2}^{\prime}(x)=x^{-1}$.

From this, immediately because $e<\pi$

$$
\frac{e}{\ln e}<\frac{\pi}{\ln \pi} \rightarrow e \cdot \ln \pi<\pi \cdot \ln e \rightarrow \ln \pi^{e}<\ln e^{\pi}
$$

and because of the monotonicity of the $\ln$ function $\pi^{e}<e^{\pi}$.