Comparing powers

Is there any way to check besides acctually calculating whether $n^x$ is $> = <$ then $m^y$?

For example how to check wheter $440902^{532446} > = < 555151^{523163}$?

• Without any restrictions on $m,n,x,y$, I don't think there's any better way other than comparing the log of both sides. – GNUSupporter 8964民主女神 地下教會 Jul 16 '18 at 22:52

2 Answers

You can make the calculation easier (or at least put it on a more eyeball-able scale) by using logarithms or other monotonic functions. Basically, you want to use a function $f$ such that $x > y$ and $f(x) > f(y)$ are equivalent.

When dealing with exponentiation, logarithms are particularly useful since you can do a lot of manipulations with them. For example:

$\log{(440902^{532446})} = 532446\log{440902}$ (which is true regardless of the base of the log)

Or possibly even more useful:

$\log_{10}{440902} = \log_{10}{100000} + \log_{10}{4.40902} = 5 + \log_{10}{4.40902}$

So the log, base 10, of the left-hand side of your question is $\approx 532446(5 + \log_{10}{4})$ while the log of the right-hand side is $\approx 523163(5 + \log_{10}{5})$. Since the logs of 4 and 5 are both going to be about 3-ish, the left-hand side has got an obvious lead.

• "The logs of $4$ and $5$ are both $3$-ish"? They're $0.602$ and $0.699$, respectively, to three digits. The difference between the actual number of digits between the two offered numbers is only $44$, which is about one part in $12$ thousand of the exponents. Approximating $4.40902$ and $5.55151$ as $4$ and $5$ yields the correct answer, but the error in doing so is about $25$ to $30$ times the actual difference, so the correct answer is attributable mostly to chance. – Brian Tung Jul 17 '18 at 0:19

Intuitively, the exponent matters much more than the base, so without computation I would guess the left side is greater. Alpha confirms this. The log of the left is over $100$ more than the log of the right, so the left is greater than the right by a factor over $e^{100} \approx 2.7\cdot 10^{43}$. To be surer, take the log of each side.