Which is bigger, $20^{25}$ or $25^{20}$ ?

Q: Which is bigger, 20^25 or 25^20?

20^25 is bigger: 20^25 = 3.35544 × 10^32 and 25^20 = 9.09494 × 10^27.

$20^{25}$ is bigger: $20^{25}$ = 3.35544 × $10^{32}$ and $25^{20}$ = 9.09494 × $10^{27}$ .

20^{25} = 3.35544\times 10^{32}\ >\ 25^{20} = 9.09494\times 10^{27}

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The curve ln x / x, which decides every comparison of a^b with b^a. 20^25 against 25^20 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: $20^{25}$ = 335544320000000000000000000000000 and $25^{20}$ = 9094947017729282379150390625.
2
So $20^{25}$ 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 20 and 25 are past e, where the function is decreasing, so the larger exponent loses and $20^{25}$ > $25^{20}$ .
$\frac{\ln 20}{20}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 20^{25} = 3.35544\times 10^{32}\ >\ 25^{20} = 9.09494\times 10^{27} \]
\[ \frac{\ln 20}{20}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

20^{25}

25^{20}

$20^{25}$ is bigger: $20^{25}$ = 3.35544 × $10^{32}$ and $25^{20}$ = 9.09494 × $10^{27}$ .

Which is bigger, 202520^{25}2025 or 252025^{20}2520?

How to work it out

Questions

Similar comparisons

Which is bigger, $20^{25}$ or $25^{20}$ ?