Which is bigger, $24^{25}$ or $25^{24}$ ?

Q: Which is bigger, 24^25 or 25^24?

24^25 is bigger: 24^25 = 3.20096 × 10^34 and 25^24 = 3.55271 × 10^33.

$24^{25}$ is bigger: $24^{25}$ = 3.20096 × $10^{34}$ and $25^{24}$ = 3.55271 × $10^{33}$ .

24^{25} = 3.20096\times 10^{34}\ >\ 25^{24} = 3.55271\times 10^{33}

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The curve ln x / x, which decides every comparison of a^b with b^a. 24^25 against 25^24 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: $24^{25}$ = 32009658644406818986777955348250624 and $25^{24}$ = 3552713678800500929355621337890625.
2
So $24^{25}$ 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 24 and 25 are past e, where the function is decreasing, so the larger exponent loses and $24^{25}$ > $25^{24}$ .
$\frac{\ln 24}{24}\ >\ \frac{\ln 25}{25}$

LaTeX source of the formulas on this page

\[ 24^{25} = 3.20096\times 10^{34}\ >\ 25^{24} = 3.55271\times 10^{33} \]
\[ \frac{\ln 24}{24}\ >\ \frac{\ln 25}{25} \]

Questions

Which is bigger,

24^{25}

25^{24}

$24^{25}$ is bigger: $24^{25}$ = 3.20096 × $10^{34}$ and $25^{24}$ = 3.55271 × $10^{33}$ .

Which is bigger, 242524^{25}2425 or 252425^{24}2524?

How to work it out

Questions

Similar comparisons

Which is bigger, $24^{25}$ or $25^{24}$ ?