Which is bigger, $20^{23}$ or $23^{20}$ ?

Q: Which is bigger, 20^23 or 23^20?

20^23 is bigger: 20^23 = 8.38860 × 10^29 and 23^20 = 1.71615 × 10^27.

$20^{23}$ is bigger: $20^{23}$ = 8.38860 × $10^{29}$ and $23^{20}$ = 1.71615 × $10^{27}$ .

20^{23} = 8.38860\times 10^{29}\ >\ 23^{20} = 1.71615\times 10^{27}

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The curve ln x / x, which decides every comparison of a^b with b^a. 20^23 against 23^20 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: $20^{23}$ = 838860800000000000000000000000 and $23^{20}$ = 1716155831334586342923895201.
2
So $20^{23}$ 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 20 and 23 are past e, where the function is decreasing, so the larger exponent loses and $20^{23}$ > $23^{20}$ .
$\frac{\ln 20}{20}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 20^{23} = 8.38860\times 10^{29}\ >\ 23^{20} = 1.71615\times 10^{27} \]
\[ \frac{\ln 20}{20}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

20^{23}

23^{20}

$20^{23}$ is bigger: $20^{23}$ = 8.38860 × $10^{29}$ and $23^{20}$ = 1.71615 × $10^{27}$ .

Which is bigger, 202320^{23}2023 or 232023^{20}2320?

How to work it out

Questions

Similar comparisons

Which is bigger, $20^{23}$ or $23^{20}$ ?