Which is bigger, $4^{13}$ or $13^{4}$ ?

Q: Which is bigger, 4^13 or 13^4?

4^13 is bigger: 4^13 = 67108864 and 13^4 = 28561.

$4^{13}$ is bigger: $4^{13}$ = 67108864 and $13^{4}$ = 28561.

4^{13} = 67108864\ >\ 13^{4} = 28561

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^13 against 13^4 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: $4^{13}$ = 67108864 and $13^{4}$ = 28561.
2
So $4^{13}$ 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 4 and 13 are past e, where the function is decreasing, so the larger exponent loses and $4^{13}$ > $13^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 13}{13}$

LaTeX source of the formulas on this page

\[ 4^{13} = 67108864\ >\ 13^{4} = 28561 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 13}{13} \]

Questions

Which is bigger,

4^{13}

13^{4}

$4^{13}$ is bigger: $4^{13}$ = 67108864 and $13^{4}$ = 28561.

Which is bigger, 4134^{13}413 or 13413^{4}134?

How to work it out

Questions

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Which is bigger, $4^{13}$ or $13^{4}$ ?