Which is bigger, $9^{22}$ or $22^{9}$ ?

Q: Which is bigger, 9^22 or 22^9?

9^22 is bigger: 9^22 = 9.84770 × 10^20 and 22^9 = 1.20726 × 10^12.

$9^{22}$ is bigger: $9^{22}$ = 9.84770 × $10^{20}$ and $22^{9}$ = 1.20726 × $10^{12}$ .

9^{22} = 9.84770\times 10^{20}\ >\ 22^{9} = 1207269217792

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The curve ln x / x, which decides every comparison of a^b with b^a. 9^22 against 22^9 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: $9^{22}$ = 984770902183611232881 and $22^{9}$ = 1207269217792.
2
So $9^{22}$ 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 9 and 22 are past e, where the function is decreasing, so the larger exponent loses and $9^{22}$ > $22^{9}$ .
$\frac{\ln 9}{9}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 9^{22} = 9.84770\times 10^{20}\ >\ 22^{9} = 1207269217792 \]
\[ \frac{\ln 9}{9}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

9^{22}

22^{9}

$9^{22}$ is bigger: $9^{22}$ = 9.84770 × $10^{20}$ and $22^{9}$ = 1.20726 × $10^{12}$ .

Which is bigger, 9229^{22}922 or 22922^{9}229?

How to work it out

Questions

Similar comparisons

Which is bigger, $9^{22}$ or $22^{9}$ ?