Which is bigger, $12^{25}$ or $25^{12}$ ?

Q: Which is bigger, 12^25 or 25^12?

12^25 is bigger: 12^25 = 9.53962 × 10^26 and 25^12 = 5.96046 × 10^16.

$12^{25}$ is bigger: $12^{25}$ = 9.53962 × $10^{26}$ and $25^{12}$ = 5.96046 × $10^{16}$ .

12^{25} = 9.53962\times 10^{26}\ >\ 25^{12} = 5.96046\times 10^{16}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^25 against 25^12 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: $12^{25}$ = 953962166440690129601298432 and $25^{12}$ = 59604644775390625.
2
So $12^{25}$ 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 12 and 25 are past e, where the function is decreasing, so the larger exponent loses and $12^{25}$ > $25^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 12^{25} = 9.53962\times 10^{26}\ >\ 25^{12} = 5.96046\times 10^{16} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

12^{25}

25^{12}

$12^{25}$ is bigger: $12^{25}$ = 9.53962 × $10^{26}$ and $25^{12}$ = 5.96046 × $10^{16}$ .

Which is bigger, 122512^{25}1225 or 251225^{12}2512?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{25}$ or $25^{12}$ ?