Which is bigger, $11^{18}$ or $18^{11}$ ?

Q: Which is bigger, 11^18 or 18^11?

11^18 is bigger: 11^18 = 5.55991 × 10^18 and 18^11 = 6.42684 × 10^13.

$11^{18}$ is bigger: $11^{18}$ = 5.55991 × $10^{18}$ and $18^{11}$ = 6.42684 × $10^{13}$ .

11^{18} = 5.55991\times 10^{18}\ >\ 18^{11} = 64268410079232

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The curve ln x / x, which decides every comparison of a^b with b^a. 11^18 against 18^11 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: $11^{18}$ = 5559917313492231481 and $18^{11}$ = 64268410079232.
2
So $11^{18}$ 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 18 are past e, where the function is decreasing, so the larger exponent loses and $11^{18}$ > $18^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 18}{18}$

LaTeX source of the formulas on this page

\[ 11^{18} = 5.55991\times 10^{18}\ >\ 18^{11} = 64268410079232 \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 18}{18} \]

Questions

Which is bigger,

11^{18}

18^{11}

$11^{18}$ is bigger: $11^{18}$ = 5.55991 × $10^{18}$ and $18^{11}$ = 6.42684 × $10^{13}$ .

Which is bigger, 111811^{18}1118 or 181118^{11}1811?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{18}$ or $18^{11}$ ?