Which is bigger, $22^{23}$ or $23^{22}$ ?

Q: Which is bigger, 22^23 or 23^22?

22^23 is bigger: 22^23 = 7.51141 × 10^30 and 23^22 = 9.07846 × 10^29.

$22^{23}$ is bigger: $22^{23}$ = 7.51141 × $10^{30}$ and $23^{22}$ = 9.07846 × $10^{29}$ .

22^{23} = 7.51141\times 10^{30}\ >\ 23^{22} = 9.07846\times 10^{29}

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The curve ln x / x, which decides every comparison of a^b with b^a. 22^23 against 23^22 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: $22^{23}$ = 7511413302012830262726227918848 and $23^{22}$ = 907846434775996175406740561329.
2
So $22^{23}$ 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 22 and 23 are past e, where the function is decreasing, so the larger exponent loses and $22^{23}$ > $23^{22}$ .
$\frac{\ln 22}{22}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 22^{23} = 7.51141\times 10^{30}\ >\ 23^{22} = 9.07846\times 10^{29} \]
\[ \frac{\ln 22}{22}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

22^{23}

23^{22}

$22^{23}$ is bigger: $22^{23}$ = 7.51141 × $10^{30}$ and $23^{22}$ = 9.07846 × $10^{29}$ .

Which is bigger, 222322^{23}2223 or 232223^{22}2322?

How to work it out

Questions

Similar comparisons

Which is bigger, $22^{23}$ or $23^{22}$ ?