How I show that $571^{61} <61^{571}$ using prime number theorem? $571$ and $61$ are primes , I want really to use some basics in number theory to show that $571^{61} <61^{571}$ without using compuation by wolfram alpha or something related to computation, I'm interested if it is possible to show that using Prime number theorem ?

The following doesn't use any aspects of prime numbers (your question leaves open whether a solution using the primality of $61$ and $571$ is necessary), but suffices to show that this isn't a particularly close inequality:
\begin{align}
61^{571} & > 32^{571} \\
         & = (2^5)^{571} \\
         & = 2^{2855} \\
         & > 2^{610} \qquad \text{like, a ginormous humongous lot larger} \\
         & = (2^{10})^{61} \\
         & = 1024^{61} \\
         & > 571^{61}
\end{align}
