Which is bigger, $13^{23}$ or $23^{13}$ ?

Q: Which is bigger, 13^23 or 23^13?

13^23 is bigger: 13^23 = 4.17539 × 10^25 and 23^13 = 5.04036 × 10^17.

$13^{23}$ is bigger: $13^{23}$ = 4.17539 × $10^{25}$ and $23^{13}$ = 5.04036 × $10^{17}$ .

13^{23} = 4.17539\times 10^{25}\ >\ 23^{13} = 5.04036\times 10^{17}

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The curve ln x / x, which decides every comparison of a^b with b^a. 13^23 against 23^13 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: $13^{23}$ = 41753905413413116367045797 and $23^{13}$ = 504036361936467383.
2
So $13^{23}$ 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 13 and 23 are past e, where the function is decreasing, so the larger exponent loses and $13^{23}$ > $23^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 13^{23} = 4.17539\times 10^{25}\ >\ 23^{13} = 5.04036\times 10^{17} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

13^{23}

23^{13}

$13^{23}$ is bigger: $13^{23}$ = 4.17539 × $10^{25}$ and $23^{13}$ = 5.04036 × $10^{17}$ .

Which is bigger, 132313^{23}1323 or 231323^{13}2313?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{23}$ or $23^{13}$ ?