Which is bigger, $11^{20}$ or $20^{11}$ ?

Q: Which is bigger, 11^20 or 20^11?

11^20 is bigger: 11^20 = 6.72749 × 10^20 and 20^11 = 2.04800 × 10^14.

$11^{20}$ is bigger: $11^{20}$ = 6.72749 × $10^{20}$ and $20^{11}$ = 2.04800 × $10^{14}$ .

11^{20} = 6.72749\times 10^{20}\ >\ 20^{11} = 2.04800\times 10^{14}

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The curve ln x / x, which decides every comparison of a^b with b^a. 11^20 against 20^11 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: $11^{20}$ = 672749994932560009201 and $20^{11}$ = 204800000000000.
2
So $11^{20}$ 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 11 and 20 are past e, where the function is decreasing, so the larger exponent loses and $11^{20}$ > $20^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 11^{20} = 6.72749\times 10^{20}\ >\ 20^{11} = 2.04800\times 10^{14} \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

11^{20}

20^{11}

$11^{20}$ is bigger: $11^{20}$ = 6.72749 × $10^{20}$ and $20^{11}$ = 2.04800 × $10^{14}$ .

Which is bigger, 112011^{20}1120 or 201120^{11}2011?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{20}$ or $20^{11}$ ?