Which is bigger, $19^{23}$ or $23^{19}$ ?

Q: Which is bigger, 19^23 or 23^19?

19^23 is bigger: 19^23 = 2.57829 × 10^29 and 23^19 = 7.46154 × 10^25.

$19^{23}$ is bigger: $19^{23}$ = 2.57829 × $10^{29}$ and $23^{19}$ = 7.46154 × $10^{25}$ .

19^{23} = 2.57829\times 10^{29}\ >\ 23^{19} = 7.46154\times 10^{25}

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The curve ln x / x, which decides every comparison of a^b with b^a. 19^23 against 23^19 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: $19^{23}$ = 257829627945307727248226067259 and $23^{19}$ = 74615470927590710561908487.
2
So $19^{23}$ 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 19 and 23 are past e, where the function is decreasing, so the larger exponent loses and $19^{23}$ > $23^{19}$ .
$\frac{\ln 19}{19}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 19^{23} = 2.57829\times 10^{29}\ >\ 23^{19} = 7.46154\times 10^{25} \]
\[ \frac{\ln 19}{19}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

19^{23}

23^{19}

$19^{23}$ is bigger: $19^{23}$ = 2.57829 × $10^{29}$ and $23^{19}$ = 7.46154 × $10^{25}$ .

Which is bigger, 192319^{23}1923 or 231923^{19}2319?

How to work it out

Questions

Similar comparisons

Which is bigger, $19^{23}$ or $23^{19}$ ?