Is there a simple way of computing when $a^n=b^m$

Question

I don't want exact equality just close enough to be useful in approximation. i.e. $2^{10 }= 10^3$ is very useful and used daily for an approximation.

Is there a do this efficiently? Is there a way to do this in my head generally?

Answer 1

Taking the logarithms,

$$n\log(a)=m\log(b),$$ or

$$\frac{\log(a)}{\log(b)}=\frac mn\in\mathbb Q.$$

So the ratio of the logarithms in the desired bases should be well approximated by a rational.

In your example

$$\frac{\log(2)}{\log(10)}=0.3010299957\cdots\approx\frac3{10}.$$

Any real number can be approximated as closely as you want by rationals using continued fractions.

Unless you are a direct descendant of Srinivasa Ramanujan, I doubt you can do it mentally. For the bases $2$ and $10$, there are no other exciting fractions.