Which is bigger, $12^{23}$ or $23^{12}$ ?

Q: Which is bigger, 12^23 or 23^12?

12^23 is bigger: 12^23 = 6.62473 × 10^24 and 23^12 = 2.19146 × 10^16.

$12^{23}$ is bigger: $12^{23}$ = 6.62473 × $10^{24}$ and $23^{12}$ = 2.19146 × $10^{16}$ .

12^{23} = 6.62473\times 10^{24}\ >\ 23^{12} = 2.19146\times 10^{16}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^23 against 23^12 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: $12^{23}$ = 6624737266949237011120128 and $23^{12}$ = 21914624432020321.
2
So $12^{23}$ 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 12 and 23 are past e, where the function is decreasing, so the larger exponent loses and $12^{23}$ > $23^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 12^{23} = 6.62473\times 10^{24}\ >\ 23^{12} = 2.19146\times 10^{16} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

12^{23}

23^{12}

$12^{23}$ is bigger: $12^{23}$ = 6.62473 × $10^{24}$ and $23^{12}$ = 2.19146 × $10^{16}$ .

Which is bigger, 122312^{23}1223 or 231223^{12}2312?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{23}$ or $23^{12}$ ?