Which is bigger, $5^{24}$ or $24^{5}$ ?

Q: Which is bigger, 5^24 or 24^5?

5^24 is bigger: 5^24 = 5.96046 × 10^16 and 24^5 = 7962624.

$5^{24}$ is bigger: $5^{24}$ = 5.96046 × $10^{16}$ and $24^{5}$ = 7962624.

5^{24} = 5.96046\times 10^{16}\ >\ 24^{5} = 7962624

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The curve ln x / x, which decides every comparison of a^b with b^a. 5^24 against 24^5 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: $5^{24}$ = 59604644775390625 and $24^{5}$ = 7962624.
2
So $5^{24}$ 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 5 and 24 are past e, where the function is decreasing, so the larger exponent loses and $5^{24}$ > $24^{5}$ .
$\frac{\ln 5}{5}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 5^{24} = 5.96046\times 10^{16}\ >\ 24^{5} = 7962624 \]
\[ \frac{\ln 5}{5}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

5^{24}

24^{5}

$5^{24}$ is bigger: $5^{24}$ = 5.96046 × $10^{16}$ and $24^{5}$ = 7962624.

Which is bigger, 5245^{24}524 or 24524^{5}245?

How to work it out

Questions

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Which is bigger, $5^{24}$ or $24^{5}$ ?