Assume \(\log_p(b)\) is rational, This means there exists integers \(m\) and \(n\) such that:
$$\begin{gather} \log_p(b) = \frac{m}{n} \\ p^{m/n} = b \\ p^m = b^n \end{gather}$$
Let \(q\) be some prime that divides \(b\). Since \(b\) is not a power of \(p\), then \(q \ne p\). Also, if \(q|b\), then \(q|b^n\), which means \(q|p^m\). A prime cannot divide another prime, so this is a contradiction. Therefore, \(\log_p(b)\) is irrational.