Which is bigger, $2^{28}$ or $28^{2}$ ?

Q: Which is bigger, 2^28 or 28^2?

2^28 is bigger: 2^28 = 268435456 and 28^2 = 784.

$2^{28}$ is bigger: $2^{28}$ = 268435456 and $28^{2}$ = 784.

2^{28} = 268435456\ >\ 28^{2} = 784

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. 2^28 against 28^2 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: $2^{28}$ = 268435456 and $28^{2}$ = 784.
2
So $2^{28}$ 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. For 2 and 28 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 2^{28} = 268435456\ >\ 28^{2} = 784 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

2^{28}

28^{2}

$2^{28}$ is bigger: $2^{28}$ = 268435456 and $28^{2}$ = 784.

Which is bigger, 2282^{28}228 or 28228^{2}282?

How to work it out

Questions

Similar comparisons

Which is bigger, $2^{28}$ or $28^{2}$ ?