Which is bigger, $20^{22}$ or $22^{20}$ ?

Q: Which is bigger, 20^22 or 22^20?

20^22 is bigger: 20^22 = 4.19430 × 10^28 and 22^20 = 7.05429 × 10^26.

$20^{22}$ is bigger: $20^{22}$ = 4.19430 × $10^{28}$ and $22^{20}$ = 7.05429 × $10^{26}$ .

20^{22} = 4.19430\times 10^{28}\ >\ 22^{20} = 7.05429\times 10^{26}

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The curve ln x / x, which decides every comparison of a^b with b^a. 20^22 against 22^20 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: $20^{22}$ = 41943040000000000000000000000 and $22^{20}$ = 705429498686404044207947776.
2
So $20^{22}$ 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 20 and 22 are past e, where the function is decreasing, so the larger exponent loses and $20^{22}$ > $22^{20}$ .
$\frac{\ln 20}{20}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 20^{22} = 4.19430\times 10^{28}\ >\ 22^{20} = 7.05429\times 10^{26} \]
\[ \frac{\ln 20}{20}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

20^{22}

22^{20}

$20^{22}$ is bigger: $20^{22}$ = 4.19430 × $10^{28}$ and $22^{20}$ = 7.05429 × $10^{26}$ .

Which is bigger, 202220^{22}2022 or 222022^{20}2220?

How to work it out

Questions

Similar comparisons

Which is bigger, $20^{22}$ or $22^{20}$ ?