Which is bigger, $4^{18}$ or $18^{4}$ ?

Q: Which is bigger, 4^18 or 18^4?

4^18 is bigger: 4^18 = 68719476736 and 18^4 = 104976.

$4^{18}$ is bigger: $4^{18}$ = 68719476736 and $18^{4}$ = 104976.

4^{18} = 68719476736\ >\ 18^{4} = 104976

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^18 against 18^4 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: $4^{18}$ = 68719476736 and $18^{4}$ = 104976.
2
So $4^{18}$ 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 18 are past e, where the function is decreasing, so the larger exponent loses and $4^{18}$ > $18^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 18}{18}$

LaTeX source of the formulas on this page

\[ 4^{18} = 68719476736\ >\ 18^{4} = 104976 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 18}{18} \]

Questions

Which is bigger,

4^{18}

18^{4}

$4^{18}$ is bigger: $4^{18}$ = 68719476736 and $18^{4}$ = 104976.

Which is bigger, 4184^{18}418 or 18418^{4}184?

How to work it out

Questions

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Which is bigger, $4^{18}$ or $18^{4}$ ?