Which is bigger, $7^{13}$ or $13^{7}$ ?

Q: Which is bigger, 7^13 or 13^7?

7^13 is bigger: 7^13 = 96889010407 and 13^7 = 62748517.

$7^{13}$ is bigger: $7^{13}$ = 96889010407 and $13^{7}$ = 62748517.

7^{13} = 96889010407\ >\ 13^{7} = 62748517

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^13 against 13^7 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: $7^{13}$ = 96889010407 and $13^{7}$ = 62748517.
2
So $7^{13}$ 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 7 and 13 are past e, where the function is decreasing, so the larger exponent loses and $7^{13}$ > $13^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 13}{13}$

LaTeX source of the formulas on this page

\[ 7^{13} = 96889010407\ >\ 13^{7} = 62748517 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 13}{13} \]

Questions

Which is bigger,

7^{13}

13^{7}

$7^{13}$ is bigger: $7^{13}$ = 96889010407 and $13^{7}$ = 62748517.

Which is bigger, 7137^{13}713 or 13713^{7}137?

How to work it out

Questions

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Which is bigger, $7^{13}$ or $13^{7}$ ?