Which is bigger, $4^{14}$ or $14^{4}$ ?

Q: Which is bigger, 4^14 or 14^4?

4^14 is bigger: 4^14 = 268435456 and 14^4 = 38416.

$4^{14}$ is bigger: $4^{14}$ = 268435456 and $14^{4}$ = 38416.

4^{14} = 268435456\ >\ 14^{4} = 38416

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^14 against 14^4 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: $4^{14}$ = 268435456 and $14^{4}$ = 38416.
2
So $4^{14}$ 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. Both 4 and 14 are past e, where the function is decreasing, so the larger exponent loses and $4^{14}$ > $14^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 14}{14}$

LaTeX source of the formulas on this page

\[ 4^{14} = 268435456\ >\ 14^{4} = 38416 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 14}{14} \]

Questions

Which is bigger,

4^{14}

14^{4}

$4^{14}$ is bigger: $4^{14}$ = 268435456 and $14^{4}$ = 38416.

Which is bigger, 4144^{14}414 or 14414^{4}144?

How to work it out

Questions

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Which is bigger, $4^{14}$ or $14^{4}$ ?