Which is bigger, $18^{22}$ or $22^{18}$ ?

Q: Which is bigger, 18^22 or 22^18?

18^22 is bigger: 18^22 = 4.13042 × 10^27 and 22^18 = 1.45749 × 10^24.

$18^{22}$ is bigger: $18^{22}$ = 4.13042 × $10^{27}$ and $22^{18}$ = 1.45749 × $10^{24}$ .

18^{22} = 4.13042\times 10^{27}\ >\ 22^{18} = 1.45749\times 10^{24}

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The curve ln x / x, which decides every comparison of a^b with b^a. 18^22 against 22^18 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: $18^{22}$ = 4130428534112329328517709824 and $22^{18}$ = 1457498964228107529355264.
2
So $18^{22}$ 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 18 and 22 are past e, where the function is decreasing, so the larger exponent loses and $18^{22}$ > $22^{18}$ .
$\frac{\ln 18}{18}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 18^{22} = 4.13042\times 10^{27}\ >\ 22^{18} = 1.45749\times 10^{24} \]
\[ \frac{\ln 18}{18}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

18^{22}

22^{18}

$18^{22}$ is bigger: $18^{22}$ = 4.13042 × $10^{27}$ and $22^{18}$ = 1.45749 × $10^{24}$ .

Which is bigger, 182218^{22}1822 or 221822^{18}2218?

How to work it out

Questions

Similar comparisons

Which is bigger, $18^{22}$ or $22^{18}$ ?