Which is bigger, $18^{23}$ or $23^{18}$ ?

Q: Which is bigger, 18^23 or 23^18?

18^23 is bigger: 18^23 = 7.43477 × 10^28 and 23^18 = 3.24415 × 10^24.

$18^{23}$ is bigger: $18^{23}$ = 7.43477 × $10^{28}$ and $23^{18}$ = 3.24415 × $10^{24}$ .

18^{23} = 7.43477\times 10^{28}\ >\ 23^{18} = 3.24415\times 10^{24}

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The curve ln x / x, which decides every comparison of a^b with b^a. 18^23 against 23^18 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: $18^{23}$ = 74347713614021927913318776832 and $23^{18}$ = 3244150909895248285300369.
2
So $18^{23}$ 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 18 and 23 are past e, where the function is decreasing, so the larger exponent loses and $18^{23}$ > $23^{18}$ .
$\frac{\ln 18}{18}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 18^{23} = 7.43477\times 10^{28}\ >\ 23^{18} = 3.24415\times 10^{24} \]
\[ \frac{\ln 18}{18}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

18^{23}

23^{18}

$18^{23}$ is bigger: $18^{23}$ = 7.43477 × $10^{28}$ and $23^{18}$ = 3.24415 × $10^{24}$ .

Which is bigger, 182318^{23}1823 or 231823^{18}2318?

How to work it out

Questions

Similar comparisons

Which is bigger, $18^{23}$ or $23^{18}$ ?