Which is bigger, $23^{30}$ or $30^{23}$ ?

Q: Which is bigger, 23^30 or 30^23?

23^30 is bigger: 23^30 = 7.10943 × 10^40 and 30^23 = 9.41431 × 10^33.

$23^{30}$ is bigger: $23^{30}$ = 7.10943 × $10^{40}$ and $30^{23}$ = 9.41431 × $10^{33}$ .

23^{30} = 7.10943\times 10^{40}\ >\ 30^{23} = 9.41431\times 10^{33}

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The curve ln x / x, which decides every comparison of a^b with b^a. 23^30 against 30^23 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: $23^{30}$ = 71094348791151363024389554286420996798449 and $30^{23}$ = 9414317882700000000000000000000000.
2
So $23^{30}$ 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 30 are past e, where the function is decreasing, so the larger exponent loses and $23^{30}$ > $30^{23}$ .
$\frac{\ln 23}{23}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 23^{30} = 7.10943\times 10^{40}\ >\ 30^{23} = 9.41431\times 10^{33} \]
\[ \frac{\ln 23}{23}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

23^{30}

30^{23}

$23^{30}$ is bigger: $23^{30}$ = 7.10943 × $10^{40}$ and $30^{23}$ = 9.41431 × $10^{33}$ .

Which is bigger, 233023^{30}2330 or 302330^{23}3023?

How to work it out

Questions

Similar comparisons

Which is bigger, $23^{30}$ or $30^{23}$ ?