Find the leftmost (most significant digits) of a large exponent calculation, say $99^{99}$

Question

I want to find the initial 10 digits of an exponent calculation whose result is a very large number - Say, $99^{99} = 3.697296 \times 10^{197}$

I only need to know the digits $3697296$

Is there any way of finding this without doing the full calculation?

Answer 1

Use common logarithms and a decent calculator. Using a fairly mediocre one on your example, I get $99\log_{10}99\approx197.5678842652$; subtracting $197$ and raising $10$ to the resulting power, I get $3.697296376497$, so the number must be about $3.697296376497\times10^{197}$.

Added: More generally, for $a^b$ calculate $b\log_{10}a$, subtract the integer part, and raise $10$ to the resulting power. If $n=\lfloor b\log_{10}a\rfloor$, the integer part of $b\log_{10}a$, your number is

$$10^{b\log_{10}a-n}\times 10^n\;,$$

and you’ll be able to read of the most significant digits from the lefthand end of $10^{b\log_{10}a-n}$.