Which is bigger, $2^{14}$ or $14^{2}$ ?

Q: Which is bigger, 2^14 or 14^2?

2^14 is bigger: 2^14 = 16384 and 14^2 = 196.

$2^{14}$ is bigger: $2^{14}$ = 16384 and $14^{2}$ = 196.

2^{14} = 16384\ >\ 14^{2} = 196

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The curve ln x / x, which decides every comparison of a^b with b^a. 2^14 against 14^2 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: $2^{14}$ = 16384 and $14^{2}$ = 196.
2
So $2^{14}$ 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 14 it depends: $2^{4}$ = $4^{2}$ are equal, $2^{3}$ < $3^{2}$ , and $2^{b}$ > $b^{2}$ for every b ≥ 5.
$\frac{\ln 2}{2}\ >\ \frac{\ln 14}{14}$

LaTeX source of the formulas on this page

\[ 2^{14} = 16384\ >\ 14^{2} = 196 \]
\[ \frac{\ln 2}{2}\ >\ \frac{\ln 14}{14} \]

Questions

Which is bigger,

2^{14}

14^{2}

$2^{14}$ is bigger: $2^{14}$ = 16384 and $14^{2}$ = 196.

Which is bigger, 2142^{14}214 or 14214^{2}142?

How to work it out

Questions

Similar comparisons

Which is bigger, $2^{14}$ or $14^{2}$ ?