Which is bigger, $11^{25}$ or $25^{11}$ ?

Q: Which is bigger, 11^25 or 25^11?

11^25 is bigger: 11^25 = 1.08347 × 10^26 and 25^11 = 2.38418 × 10^15.

$11^{25}$ is bigger: $11^{25}$ = 1.08347 × $10^{26}$ and $25^{11}$ = 2.38418 × $10^{15}$ .

11^{25} = 1.08347\times 10^{26}\ >\ 25^{11} = 2.38418\times 10^{15}

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The curve ln x / x, which decides every comparison of a^b with b^a. 11^25 against 25^11 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: $11^{25}$ = 108347059433883722041830251 and $25^{11}$ = 2384185791015625.
2
So $11^{25}$ 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 11 and 25 are past e, where the function is decreasing, so the larger exponent loses and $11^{25}$ > $25^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 11^{25} = 1.08347\times 10^{26}\ >\ 25^{11} = 2.38418\times 10^{15} \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

11^{25}

25^{11}

$11^{25}$ is bigger: $11^{25}$ = 1.08347 × $10^{26}$ and $25^{11}$ = 2.38418 × $10^{15}$ .

Which is bigger, 112511^{25}1125 or 251125^{11}2511?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{25}$ or $25^{11}$ ?