2022-01-22

Comparing Powers of Different Bases
How can I know if one power is bigger than the other when the bases are different?
For example, considering ${2}^{10}$ and ${10}^{3}$ the former is the greater one, but how to prove this? Logarithms? I'll be working with big numbers, and though a more general solution is really appreciated, I will be comparing exactly powers of $2$ and $10$.

Tyrn7i

Expert

${\mathrm{log}}_{2}$ is the way to go.
${\mathrm{log}}_{2}\left({2}^{2000}\right)=2000$
${\mathrm{log}}_{2}\left({10}^{800}\right)=800{\mathrm{log}}_{2}\left(10\right)$
So which is bigger $20$ or $8{\mathrm{log}}_{2}\left(10\right)$?
Let's see $20$ is smaller than $8×3=24$ and
${\mathrm{log}}_{2}\left(10\right)>{\mathrm{log}}_{2}\left(8\right)={\mathrm{log}}_{2}\left({2}^{3}\right)=3$. So it looks like:
${2}^{2000}<{10}^{800}$
(no calculator required).
To compare:
$3={\mathrm{log}}_{2}\left({2}^{3}\right)={\mathrm{log}}_{2}\left(8\right)<{\mathrm{log}}_{2}\left(10\right)<{\mathrm{log}}_{2}\left(16\right)={\mathrm{log}}_{2}\left({2}^{4}\right)=4$

Do you have a similar question?