Which is bigger, $14^{20}$ or $20^{14}$ ?

Q: Which is bigger, 14^20 or 20^14?

14^20 is bigger: 14^20 = 8.36682 × 10^22 and 20^14 = 1.63840 × 10^18.

$14^{20}$ is bigger: $14^{20}$ = 8.36682 × $10^{22}$ and $20^{14}$ = 1.63840 × $10^{18}$ .

14^{20} = 8.36682\times 10^{22}\ >\ 20^{14} = 1.63840\times 10^{18}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^20 against 20^14 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: $14^{20}$ = 83668255425284801560576 and $20^{14}$ = 1638400000000000000.
2
So $14^{20}$ 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 14 and 20 are past e, where the function is decreasing, so the larger exponent loses and $14^{20}$ > $20^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 14^{20} = 8.36682\times 10^{22}\ >\ 20^{14} = 1.63840\times 10^{18} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

14^{20}

20^{14}

$14^{20}$ is bigger: $14^{20}$ = 8.36682 × $10^{22}$ and $20^{14}$ = 1.63840 × $10^{18}$ .

Which is bigger, 142014^{20}1420 or 201420^{14}2014?

How to work it out

Questions

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Which is bigger, $14^{20}$ or $20^{14}$ ?