Which is bigger, $11^{22}$ or $22^{11}$ ?

Q: Which is bigger, 11^22 or 22^11?

11^22 is bigger: 11^22 = 8.14027 × 10^22 and 22^11 = 5.84318 × 10^14.

$11^{22}$ is bigger: $11^{22}$ = 8.14027 × $10^{22}$ and $22^{11}$ = 5.84318 × $10^{14}$ .

11^{22} = 8.14027\times 10^{22}\ >\ 22^{11} = 5.84318\times 10^{14}

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The curve ln x / x, which decides every comparison of a^b with b^a. 11^22 against 22^11 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: $11^{22}$ = 81402749386839761113321 and $22^{11}$ = 584318301411328.
2
So $11^{22}$ 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 22 are past e, where the function is decreasing, so the larger exponent loses and $11^{22}$ > $22^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 11^{22} = 8.14027\times 10^{22}\ >\ 22^{11} = 5.84318\times 10^{14} \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

11^{22}

22^{11}

$11^{22}$ is bigger: $11^{22}$ = 8.14027 × $10^{22}$ and $22^{11}$ = 5.84318 × $10^{14}$ .

Which is bigger, 112211^{22}1122 or 221122^{11}2211?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{22}$ or $22^{11}$ ?