Which is bigger, $11^{21}$ or $21^{11}$ ?

Q: Which is bigger, 11^21 or 21^11?

11^21 is bigger: 11^21 = 7.40024 × 10^21 and 21^11 = 3.50277 × 10^14.

$11^{21}$ is bigger: $11^{21}$ = 7.40024 × $10^{21}$ and $21^{11}$ = 3.50277 × $10^{14}$ .

11^{21} = 7.40024\times 10^{21}\ >\ 21^{11} = 3.50277\times 10^{14}

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The curve ln x / x, which decides every comparison of a^b with b^a. 11^21 against 21^11 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: $11^{21}$ = 7400249944258160101211 and $21^{11}$ = 350277500542221.
2
So $11^{21}$ 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 21 are past e, where the function is decreasing, so the larger exponent loses and $11^{21}$ > $21^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 21}{21}$

LaTeX source of the formulas on this page

\[ 11^{21} = 7.40024\times 10^{21}\ >\ 21^{11} = 3.50277\times 10^{14} \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger,

11^{21}

21^{11}

$11^{21}$ is bigger: $11^{21}$ = 7.40024 × $10^{21}$ and $21^{11}$ = 3.50277 × $10^{14}$ .

Which is bigger, 112111^{21}1121 or 211121^{11}2111?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{21}$ or $21^{11}$ ?