Which is bigger, $4^{23}$ or $23^{4}$ ?

Q: Which is bigger, 4^23 or 23^4?

4^23 is bigger: 4^23 = 7.03687 × 10^13 and 23^4 = 279841.

$4^{23}$ is bigger: $4^{23}$ = 7.03687 × $10^{13}$ and $23^{4}$ = 279841.

4^{23} = 70368744177664\ >\ 23^{4} = 279841

Explain it step by step in chat Make a short video of it

The curve ln x / x, which decides every comparison of a^b with b^a. 4^23 against 23^4 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: $4^{23}$ = 70368744177664 and $23^{4}$ = 279841.
2
So $4^{23}$ 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 4 and 23 are past e, where the function is decreasing, so the larger exponent loses and $4^{23}$ > $23^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 4^{23} = 70368744177664\ >\ 23^{4} = 279841 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

4^{23}

23^{4}

$4^{23}$ is bigger: $4^{23}$ = 7.03687 × $10^{13}$ and $23^{4}$ = 279841.

Which is bigger, 4234^{23}423 or 23423^{4}234?

How to work it out

Questions

Similar comparisons

Which is bigger, $4^{23}$ or $23^{4}$ ?