Which is bigger, $2^{4}$ or $4^{2}$ ?

Q: Which is bigger, 2^4 or 4^2?

2^4 and 4^2 are equal: both are 16.

$2^{4}$ and $4^{2}$ are equal: both are 16.

2^{4} = 16\ =\ 4^{2} = 16

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. 2^4 against 4^2 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: $2^{4}$ = 16 and $4^{2}$ = 16.
2
So equal.
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. For 2 and 4 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ =\ \frac{\ln 4}{4}$

LaTeX source of the formulas on this page

\[ 2^{4} = 16\ =\ 4^{2} = 16 \]
\[ \frac{\ln 2}{2}\ =\ \frac{\ln 4}{4} \]

Questions

Which is bigger,

2^{4}

4^{2}

$2^{4}$ and $4^{2}$ are equal: both are 16.

Similar comparisons

$2^{5}$ or $5^{2}$ ?
$3^{4}$ or $4^{3}$ ?
$2^{3}$ or $3^{2}$ ?
$e^{\pi}$ or $\pi^{e}$ ?

Which is bigger, 242^{4}24 or 424^{2}42?

How to work it out

Questions

Similar comparisons

Which is bigger, $2^{4}$ or $4^{2}$ ?