Which is bigger, $7^{24}$ or $24^{7}$ ?

Q: Which is bigger, 7^24 or 24^7?

7^24 is bigger: 7^24 = 1.91581 × 10^20 and 24^7 = 4586471424.

$7^{24}$ is bigger: $7^{24}$ = 1.91581 × $10^{20}$ and $24^{7}$ = 4586471424.

7^{24} = 1.91581\times 10^{20}\ >\ 24^{7} = 4586471424

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^24 against 24^7 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: $7^{24}$ = 191581231380566414401 and $24^{7}$ = 4586471424.
2
So $7^{24}$ 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 7 and 24 are past e, where the function is decreasing, so the larger exponent loses and $7^{24}$ > $24^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 7^{24} = 1.91581\times 10^{20}\ >\ 24^{7} = 4586471424 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

7^{24}

24^{7}

$7^{24}$ is bigger: $7^{24}$ = 1.91581 × $10^{20}$ and $24^{7}$ = 4586471424.

Which is bigger, 7247^{24}724 or 24724^{7}247?

How to work it out

Questions

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Which is bigger, $7^{24}$ or $24^{7}$ ?