Which is bigger, $18^{24}$ or $24^{18}$ ?

Q: Which is bigger, 18^24 or 24^18?

18^24 is bigger: 18^24 = 1.33825 × 10^30 and 24^18 = 6.97914 × 10^24.

$18^{24}$ is bigger: $18^{24}$ = 1.33825 × $10^{30}$ and $24^{18}$ = 6.97914 × $10^{24}$ .

18^{24} = 1.33825\times 10^{30}\ >\ 24^{18} = 6.97914\times 10^{24}

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The curve ln x / x, which decides every comparison of a^b with b^a. 18^24 against 24^18 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: $18^{24}$ = 1338258845052394702439737982976 and $24^{18}$ = 6979147079584381377970176.
2
So $18^{24}$ 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 18 and 24 are past e, where the function is decreasing, so the larger exponent loses and $18^{24}$ > $24^{18}$ .
$\frac{\ln 18}{18}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 18^{24} = 1.33825\times 10^{30}\ >\ 24^{18} = 6.97914\times 10^{24} \]
\[ \frac{\ln 18}{18}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

18^{24}

24^{18}

$18^{24}$ is bigger: $18^{24}$ = 1.33825 × $10^{30}$ and $24^{18}$ = 6.97914 × $10^{24}$ .

Which is bigger, 182418^{24}1824 or 241824^{18}2418?

How to work it out

Questions

Similar comparisons

Which is bigger, $18^{24}$ or $24^{18}$ ?