Which is bigger, $13^{20}$ or $20^{13}$ ?

Q: Which is bigger, 13^20 or 20^13?

13^20 is bigger: 13^20 = 1.90049 × 10^22 and 20^13 = 8.19200 × 10^16.

$13^{20}$ is bigger: $13^{20}$ = 1.90049 × $10^{22}$ and $20^{13}$ = 8.19200 × $10^{16}$ .

13^{20} = 1.90049\times 10^{22}\ >\ 20^{13} = 8.19200\times 10^{16}

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The curve ln x / x, which decides every comparison of a^b with b^a. 13^20 against 20^13 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: $13^{20}$ = 19004963774880799438801 and $20^{13}$ = 81920000000000000.
2
So $13^{20}$ 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 20 are past e, where the function is decreasing, so the larger exponent loses and $13^{20}$ > $20^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 13^{20} = 1.90049\times 10^{22}\ >\ 20^{13} = 8.19200\times 10^{16} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

13^{20}

20^{13}

$13^{20}$ is bigger: $13^{20}$ = 1.90049 × $10^{22}$ and $20^{13}$ = 8.19200 × $10^{16}$ .

Which is bigger, 132013^{20}1320 or 201320^{13}2013?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{20}$ or $20^{13}$ ?