How to calculate large exponents by hand?

Question

How to calculate large exponents by hand like they did in ancient times? Is it something to do with Prosthaphaeresis? for example calculate $2^{15}$.

Answer 1

Use logarithms perhaps?

$$\log_{10} 2^{15} = 15\log_{10} 2 \sim 15\cdot0.3 =4.5$$ So that: $$2^{15} \sim 10^{4.5} = 10^4\sqrt{10} \sim3\cdot 10^4$$ Which is just 7% percent away, and can be done in your head as long as you remember that $\log_{10} 2 \sim 0.3$, which most engineers probably do.