Which is bigger, $8^{24}$ or $24^{8}$ ?

Q: Which is bigger, 8^24 or 24^8?

8^24 is bigger: 8^24 = 4.72236 × 10^21 and 24^8 = 110075314176.

$8^{24}$ is bigger: $8^{24}$ = 4.72236 × $10^{21}$ and $24^{8}$ = 110075314176.

8^{24} = 4.72236\times 10^{21}\ >\ 24^{8} = 110075314176

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The curve ln x / x, which decides every comparison of a^b with b^a. 8^24 against 24^8 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: $8^{24}$ = 4722366482869645213696 and $24^{8}$ = 110075314176.
2
So $8^{24}$ 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 8 and 24 are past e, where the function is decreasing, so the larger exponent loses and $8^{24}$ > $24^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 8^{24} = 4.72236\times 10^{21}\ >\ 24^{8} = 110075314176 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

8^{24}

24^{8}

$8^{24}$ is bigger: $8^{24}$ = 4.72236 × $10^{21}$ and $24^{8}$ = 110075314176.

Which is bigger, 8248^{24}824 or 24824^{8}248?

How to work it out

Questions

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Which is bigger, $8^{24}$ or $24^{8}$ ?