Which is bigger, $6^{23}$ or $23^{6}$ ?

Q: Which is bigger, 6^23 or 23^6?

6^23 is bigger: 6^23 = 7.89730 × 10^17 and 23^6 = 148035889.

$6^{23}$ is bigger: $6^{23}$ = 7.89730 × $10^{17}$ and $23^{6}$ = 148035889.

6^{23} = 7.89730\times 10^{17}\ >\ 23^{6} = 148035889

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^23 against 23^6 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: $6^{23}$ = 789730223053602816 and $23^{6}$ = 148035889.
2
So $6^{23}$ 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 6 and 23 are past e, where the function is decreasing, so the larger exponent loses and $6^{23}$ > $23^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 6^{23} = 7.89730\times 10^{17}\ >\ 23^{6} = 148035889 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

6^{23}

23^{6}

$6^{23}$ is bigger: $6^{23}$ = 7.89730 × $10^{17}$ and $23^{6}$ = 148035889.

Which is bigger, 6236^{23}623 or 23623^{6}236?

How to work it out

Questions

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Which is bigger, $6^{23}$ or $23^{6}$ ?