How to derive order of magnitude when given any base raised to some exponent?

Question

How does one quickly derive order of magnitude without needing to compute the value and count digits?

Say you’re given $26^{25}$ which has order of magnitude $10^{35}$.

How does one get from an exponent of $25$ to $35$ (in a different base)?

Answer 1

You can use logarithms:$$26^{25} = 10^x \implies 25\log_{10} 26 = x\implies x \approx 35.37.$$ This tells us that $26^{25}$ has order of magnitude $10^{35}$.