Which is bigger, $23^{24}$ or $24^{23}$ ?

Q: Which is bigger, 23^24 or 24^23?

23^24 is bigger: 23^24 = 4.80250 × 10^32 and 24^23 = 5.55723 × 10^31.

$23^{24}$ is bigger: $23^{24}$ = 4.80250 × $10^{32}$ and $24^{23}$ = 5.55723 × $10^{31}$ .

23^{24} = 4.80250\times 10^{32}\ >\ 24^{23} = 5.55723\times 10^{31}

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The curve ln x / x, which decides every comparison of a^b with b^a. 23^24 against 24^23 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: $23^{24}$ = 480250763996501976790165756943041 and $24^{23}$ = 55572324035428505185378394701824.
2
So $23^{24}$ 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 24 are past e, where the function is decreasing, so the larger exponent loses and $23^{24}$ > $24^{23}$ .
$\frac{\ln 23}{23}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 23^{24} = 4.80250\times 10^{32}\ >\ 24^{23} = 5.55723\times 10^{31} \]
\[ \frac{\ln 23}{23}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

23^{24}

24^{23}

$23^{24}$ is bigger: $23^{24}$ = 4.80250 × $10^{32}$ and $24^{23}$ = 5.55723 × $10^{31}$ .

Which is bigger, 232423^{24}2324 or 242324^{23}2423?

How to work it out

Questions

Similar comparisons

Which is bigger, $23^{24}$ or $24^{23}$ ?