Which is bigger, $2^{13}$ or $13^{2}$ ?

Q: Which is bigger, 2^13 or 13^2?

2^13 is bigger: 2^13 = 8192 and 13^2 = 169.

$2^{13}$ is bigger: $2^{13}$ = 8192 and $13^{2}$ = 169.

2^{13} = 8192\ >\ 13^{2} = 169

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^13 against 13^2 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: $2^{13}$ = 8192 and $13^{2}$ = 169.
2
So $2^{13}$ 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 13 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 13}{13}$

LaTeX source of the formulas on this page

\[ 2^{13} = 8192\ >\ 13^{2} = 169 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 13}{13} \]

Questions

Which is bigger,

2^{13}

13^{2}

$2^{13}$ is bigger: $2^{13}$ = 8192 and $13^{2}$ = 169.

Which is bigger, 2132^{13}213 or 13213^{2}132?

How to work it out

Questions

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Which is bigger, $2^{13}$ or $13^{2}$ ?