Which is bigger, $17^{23}$ or $23^{17}$ ?

Q: Which is bigger, 17^23 or 23^17?

17^23 is bigger: 17^23 = 1.99675 × 10^28 and 23^17 = 1.41050 × 10^23.

$17^{23}$ is bigger: $17^{23}$ = 1.99675 × $10^{28}$ and $23^{17}$ = 1.41050 × $10^{23}$ .

17^{23} = 1.99675\times 10^{28}\ >\ 23^{17} = 1.41050\times 10^{23}

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The curve ln x / x, which decides every comparison of a^b with b^a. 17^23 against 23^17 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: $17^{23}$ = 19967568900859523802559065713 and $23^{17}$ = 141050039560662968926103.
2
So $17^{23}$ 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 17 and 23 are past e, where the function is decreasing, so the larger exponent loses and $17^{23}$ > $23^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 17^{23} = 1.99675\times 10^{28}\ >\ 23^{17} = 1.41050\times 10^{23} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

17^{23}

23^{17}

$17^{23}$ is bigger: $17^{23}$ = 1.99675 × $10^{28}$ and $23^{17}$ = 1.41050 × $10^{23}$ .

Which is bigger, 172317^{23}1723 or 231723^{17}2317?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{23}$ or $23^{17}$ ?