Which is bigger, $9^{24}$ or $24^{9}$ ?

Q: Which is bigger, 9^24 or 24^9?

9^24 is bigger: 9^24 = 7.97664 × 10^22 and 24^9 = 2.64180 × 10^12.

$9^{24}$ is bigger: $9^{24}$ = 7.97664 × $10^{22}$ and $24^{9}$ = 2.64180 × $10^{12}$ .

9^{24} = 7.97664\times 10^{22}\ >\ 24^{9} = 2641807540224

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The curve ln x / x, which decides every comparison of a^b with b^a. 9^24 against 24^9 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: $9^{24}$ = 79766443076872509863361 and $24^{9}$ = 2641807540224.
2
So $9^{24}$ 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 9 and 24 are past e, where the function is decreasing, so the larger exponent loses and $9^{24}$ > $24^{9}$ .
$\frac{\ln 9}{9}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 9^{24} = 7.97664\times 10^{22}\ >\ 24^{9} = 2641807540224 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

9^{24}

24^{9}

$9^{24}$ is bigger: $9^{24}$ = 7.97664 × $10^{22}$ and $24^{9}$ = 2.64180 × $10^{12}$ .

Which is bigger, 9249^{24}924 or 24924^{9}249?

How to work it out

Questions

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Which is bigger, $9^{24}$ or $24^{9}$ ?