Which is bigger, $13^{17}$ or $17^{13}$ ?

Q: Which is bigger, 13^17 or 17^13?

13^17 is bigger: 13^17 = 8.65041 × 10^18 and 17^13 = 9.90457 × 10^15.

$13^{17}$ is bigger: $13^{17}$ = 8.65041 × $10^{18}$ and $17^{13}$ = 9.90457 × $10^{15}$ .

13^{17} = 8.65041\times 10^{18}\ >\ 17^{13} = 9.90457\times 10^{15}

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The curve ln x / x, which decides every comparison of a^b with b^a. 13^17 against 17^13 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: $13^{17}$ = 8650415919381337933 and $17^{13}$ = 9904578032905937.
2
So $13^{17}$ 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 13 and 17 are past e, where the function is decreasing, so the larger exponent loses and $13^{17}$ > $17^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 17}{17}$

LaTeX source of the formulas on this page

\[ 13^{17} = 8.65041\times 10^{18}\ >\ 17^{13} = 9.90457\times 10^{15} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 17}{17} \]

Questions

Which is bigger,

13^{17}

17^{13}

$13^{17}$ is bigger: $13^{17}$ = 8.65041 × $10^{18}$ and $17^{13}$ = 9.90457 × $10^{15}$ .

Which is bigger, 131713^{17}1317 or 171317^{13}1713?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{17}$ or $17^{13}$ ?