Which is bigger, $13^{24}$ or $24^{13}$ ?

Q: Which is bigger, 13^24 or 24^13?

13^24 is bigger: 13^24 = 5.42800 × 10^26 and 24^13 = 8.76488 × 10^17.

$13^{24}$ is bigger: $13^{24}$ = 5.42800 × $10^{26}$ and $24^{13}$ = 8.76488 × $10^{17}$ .

13^{24} = 5.42800\times 10^{26}\ >\ 24^{13} = 8.76488\times 10^{17}

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The curve ln x / x, which decides every comparison of a^b with b^a. 13^24 against 24^13 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: $13^{24}$ = 542800770374370512771595361 and $24^{13}$ = 876488338465357824.
2
So $13^{24}$ 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 13 and 24 are past e, where the function is decreasing, so the larger exponent loses and $13^{24}$ > $24^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 13^{24} = 5.42800\times 10^{26}\ >\ 24^{13} = 8.76488\times 10^{17} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

13^{24}

24^{13}

$13^{24}$ is bigger: $13^{24}$ = 5.42800 × $10^{26}$ and $24^{13}$ = 8.76488 × $10^{17}$ .

Which is bigger, 132413^{24}1324 or 241324^{13}2413?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{24}$ or $24^{13}$ ?