Which is bigger, $13^{25}$ or $25^{13}$ ?

Q: Which is bigger, 13^25 or 25^13?

13^25 is bigger: 13^25 = 7.05641 × 10^27 and 25^13 = 1.49011 × 10^18.

$13^{25}$ is bigger: $13^{25}$ = 7.05641 × $10^{27}$ and $25^{13}$ = 1.49011 × $10^{18}$ .

13^{25} = 7.05641\times 10^{27}\ >\ 25^{13} = 1.49011\times 10^{18}

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The curve ln x / x, which decides every comparison of a^b with b^a. 13^25 against 25^13 is the same question as which of the two marked points is higher; the curve peaks at x = e.

How to work it out

1
Compute both exactly: $13^{25}$ = 7056410014866816666030739693 and $25^{13}$ = 1490116119384765625.
2
So $13^{25}$ is bigger.
3
The reason: taking logs, $a^{b}$ against $b^{a}$ is the same as $\frac{\ln a}{a}$ against $\frac{\ln b}{b}$ . The function $\frac{\ln x}{x}$ rises until $x = e \approx 2.718$ and falls afterwards. Both 13 and 25 are past e, where the function is decreasing, so the larger exponent loses and $13^{25}$ > $25^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 13^{25} = 7.05641\times 10^{27}\ >\ 25^{13} = 1.49011\times 10^{18} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

13^{25}

25^{13}

$13^{25}$ is bigger: $13^{25}$ = 7.05641 × $10^{27}$ and $25^{13}$ = 1.49011 × $10^{18}$ .

Which is bigger, 132513^{25}1325 or 251325^{13}2513?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{25}$ or $25^{13}$ ?