Which is bigger, $8^{13}$ or $13^{8}$ ?

Q: Which is bigger, 8^13 or 13^8?

8^13 is bigger: 8^13 = 549755813888 and 13^8 = 815730721.

$8^{13}$ is bigger: $8^{13}$ = 549755813888 and $13^{8}$ = 815730721.

8^{13} = 549755813888\ >\ 13^{8} = 815730721

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The curve ln x / x, which decides every comparison of a^b with b^a. 8^13 against 13^8 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: $8^{13}$ = 549755813888 and $13^{8}$ = 815730721.
2
So $8^{13}$ 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 13 are past e, where the function is decreasing, so the larger exponent loses and $8^{13}$ > $13^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 13}{13}$

LaTeX source of the formulas on this page

\[ 8^{13} = 549755813888\ >\ 13^{8} = 815730721 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 13}{13} \]

Questions

Which is bigger,

8^{13}

13^{8}

$8^{13}$ is bigger: $8^{13}$ = 549755813888 and $13^{8}$ = 815730721.

Which is bigger, 8138^{13}813 or 13813^{8}138?

How to work it out

Questions

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Which is bigger, $8^{13}$ or $13^{8}$ ?