Which is bigger, $23^{29}$ or $29^{23}$ ?

Q: Which is bigger, 23^29 or 29^23?

23^29 is bigger: 23^29 = 3.09105 × 10^39 and 29^23 = 4.31672 × 10^33.

$23^{29}$ is bigger: $23^{29}$ = 3.09105 × $10^{39}$ and $29^{23}$ = 4.31672 × $10^{33}$ .

23^{29} = 3.09105\times 10^{39}\ >\ 29^{23} = 4.31672\times 10^{33}

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The curve ln x / x, which decides every comparison of a^b with b^a. 23^29 against 29^23 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: $23^{29}$ = 3091058643093537522799545838540043339063 and $29^{23}$ = 4316720717749415770740818372739989.
2
So $23^{29}$ 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 29 are past e, where the function is decreasing, so the larger exponent loses and $23^{29}$ > $29^{23}$ .
$\frac{\ln 23}{23}\ >\ \frac{\ln 29}{29}$

LaTeX source of the formulas on this page

\[ 23^{29} = 3.09105\times 10^{39}\ >\ 29^{23} = 4.31672\times 10^{33} \]
\[ \frac{\ln 23}{23}\ >\ \frac{\ln 29}{29} \]

Questions

Which is bigger,

23^{29}

29^{23}

$23^{29}$ is bigger: $23^{29}$ = 3.09105 × $10^{39}$ and $29^{23}$ = 4.31672 × $10^{33}$ .

Which is bigger, 232923^{29}2329 or 292329^{23}2923?

How to work it out

Questions

Similar comparisons

Which is bigger, $23^{29}$ or $29^{23}$ ?