Which is bigger, $14^{23}$ or $23^{14}$ ?

Q: Which is bigger, 14^23 or 23^14?

14^23 is bigger: 14^23 = 2.29585 × 10^26 and 23^14 = 1.15928 × 10^19.

$14^{23}$ is bigger: $14^{23}$ = 2.29585 × $10^{26}$ and $23^{14}$ = 1.15928 × $10^{19}$ .

14^{23} = 2.29585\times 10^{26}\ >\ 23^{14} = 1.15928\times 10^{19}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^23 against 23^14 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: $14^{23}$ = 229585692886981495482220544 and $23^{14}$ = 11592836324538749809.
2
So $14^{23}$ 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 14 and 23 are past e, where the function is decreasing, so the larger exponent loses and $14^{23}$ > $23^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 14^{23} = 2.29585\times 10^{26}\ >\ 23^{14} = 1.15928\times 10^{19} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

14^{23}

23^{14}

$14^{23}$ is bigger: $14^{23}$ = 2.29585 × $10^{26}$ and $23^{14}$ = 1.15928 × $10^{19}$ .

Which is bigger, 142314^{23}1423 or 231423^{14}2314?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{23}$ or $23^{14}$ ?