How many digits will $5^{4^{3^2}}$ produce?

Question

According to Wolfram Alpha, $5^{4^{3^2}}$ evaluates to an integer with $183\,231$ digits. How does one find out how many digits such a calculation will produce?

Answer 1

A number $n \in \mathbf N^+$ has $\lfloor\log_{10}n\rfloor+1$ digits. For $n = 5^{4^{3^2}}$, we have \begin{align*} \log_{10} 5^{4^{3^2}} &= 4^9 \cdot \log_{10} 5\\ &= 262\,144 \cdot \log_{10} 5\\ &\approx 183\,230.8 \end{align*} So $n$ has $183\,231$ digits.