Which is bigger, $2^{23}$ or $23^{2}$ ?

Q: Which is bigger, 2^23 or 23^2?

2^23 is bigger: 2^23 = 8388608 and 23^2 = 529.

$2^{23}$ is bigger: $2^{23}$ = 8388608 and $23^{2}$ = 529.

2^{23} = 8388608\ >\ 23^{2} = 529

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^23 against 23^2 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: $2^{23}$ = 8388608 and $23^{2}$ = 529.
2
So $2^{23}$ 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. For 2 and 23 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 23}{23}$

LaTeX source of the formulas on this page

\[ 2^{23} = 8388608\ >\ 23^{2} = 529 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 23}{23} \]

Questions

Which is bigger,

2^{23}

23^{2}

$2^{23}$ is bigger: $2^{23}$ = 8388608 and $23^{2}$ = 529.

Which is bigger, 2232^{23}223 or 23223^{2}232?

How to work it out

Questions

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Which is bigger, $2^{23}$ or $23^{2}$ ?