Which is bigger, $15^{25}$ or $25^{15}$ ?

Q: Which is bigger, 15^25 or 25^15?

15^25 is bigger: 15^25 = 2.52511 × 10^29 and 25^15 = 9.31322 × 10^20.

$15^{25}$ is bigger: $15^{25}$ = 2.52511 × $10^{29}$ and $25^{15}$ = 9.31322 × $10^{20}$ .

15^{25} = 2.52511\times 10^{29}\ >\ 25^{15} = 9.31322\times 10^{20}

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The curve ln x / x, which decides every comparison of a^b with b^a. 15^25 against 25^15 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: $15^{25}$ = 252511682940423488616943359375 and $25^{15}$ = 931322574615478515625.
2
So $15^{25}$ 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 15 and 25 are past e, where the function is decreasing, so the larger exponent loses and $15^{25}$ > $25^{15}$ .
$\frac{\ln 15}{15}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 15^{25} = 2.52511\times 10^{29}\ >\ 25^{15} = 9.31322\times 10^{20} \]
\[ \frac{\ln 15}{15}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

15^{25}

25^{15}

$15^{25}$ is bigger: $15^{25}$ = 2.52511 × $10^{29}$ and $25^{15}$ = 9.31322 × $10^{20}$ .

Which is bigger, 152515^{25}1525 or 251525^{15}2515?

How to work it out

Questions

Similar comparisons

Which is bigger, $15^{25}$ or $25^{15}$ ?