Which is bigger, $21^{23}$ or $23^{21}$ ?

Q: Which is bigger, 21^23 or 23^21?

21^23 is bigger: 21^23 = 2.57658 × 10^30 and 23^21 = 3.94715 × 10^28.

$21^{23}$ is bigger: $21^{23}$ = 2.57658 × $10^{30}$ and $23^{21}$ = 3.94715 × $10^{28}$ .

21^{23} = 2.57658\times 10^{30}\ >\ 23^{21} = 3.94715\times 10^{28}

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The curve ln x / x, which decides every comparison of a^b with b^a. 21^23 against 23^21 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: $21^{23}$ = 2576580875108218291929075869661 and $23^{21}$ = 39471584120695485887249589623.
2
So $21^{23}$ 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 21 and 23 are past e, where the function is decreasing, so the larger exponent loses and $21^{23}$ > $23^{21}$ .
$\frac{\ln 21}{21}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 21^{23} = 2.57658\times 10^{30}\ >\ 23^{21} = 3.94715\times 10^{28} \]
\[ \frac{\ln 21}{21}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

21^{23}

23^{21}

$21^{23}$ is bigger: $21^{23}$ = 2.57658 × $10^{30}$ and $23^{21}$ = 3.94715 × $10^{28}$ .

Which is bigger, 212321^{23}2123 or 232123^{21}2321?

How to work it out

Questions

Similar comparisons

Which is bigger, $21^{23}$ or $23^{21}$ ?