Which is bigger, $23^{28}$ or $28^{23}$ ?

Q: Which is bigger, 23^28 or 28^23?

23^28 is bigger: 23^28 = 1.34393 × 10^38 and 28^23 = 1.92590 × 10^33.

$23^{28}$ is bigger: $23^{28}$ = 1.34393 × $10^{38}$ and $28^{23}$ = 1.92590 × $10^{33}$ .

23^{28} = 1.34393\times 10^{38}\ >\ 28^{23} = 1.92590\times 10^{33}

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The curve ln x / x, which decides every comparison of a^b with b^a. 23^28 against 28^23 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: $23^{28}$ = 134393854047545109686936775588697536481 and $28^{23}$ = 1925904380037276068854119113162752.
2
So $23^{28}$ 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 23 and 28 are past e, where the function is decreasing, so the larger exponent loses and $23^{28}$ > $28^{23}$ .
$\frac{\ln 23}{23}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 23^{28} = 1.34393\times 10^{38}\ >\ 28^{23} = 1.92590\times 10^{33} \]
\[ \frac{\ln 23}{23}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

23^{28}

28^{23}

$23^{28}$ is bigger: $23^{28}$ = 1.34393 × $10^{38}$ and $28^{23}$ = 1.92590 × $10^{33}$ .

Which is bigger, 232823^{28}2328 or 282328^{23}2823?

How to work it out

Questions

Similar comparisons

Which is bigger, $23^{28}$ or $28^{23}$ ?