Which is bigger, $4^{20}$ or $20^{4}$ ?

Q: Which is bigger, 4^20 or 20^4?

4^20 is bigger: 4^20 = 1.09951 × 10^12 and 20^4 = 160000.

$4^{20}$ is bigger: $4^{20}$ = 1.09951 × $10^{12}$ and $20^{4}$ = 160000.

4^{20} = 1099511627776\ >\ 20^{4} = 160000

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^20 against 20^4 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: $4^{20}$ = 1099511627776 and $20^{4}$ = 160000.
2
So $4^{20}$ 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 4 and 20 are past e, where the function is decreasing, so the larger exponent loses and $4^{20}$ > $20^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 4^{20} = 1099511627776\ >\ 20^{4} = 160000 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

4^{20}

20^{4}

$4^{20}$ is bigger: $4^{20}$ = 1.09951 × $10^{12}$ and $20^{4}$ = 160000.

Which is bigger, 4204^{20}420 or 20420^{4}204?

How to work it out

Questions

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Which is bigger, $4^{20}$ or $20^{4}$ ?