Which is bigger, $6^{18}$ or $18^{6}$ ?

Q: Which is bigger, 6^18 or 18^6?

6^18 is bigger: 6^18 = 1.01559 × 10^14 and 18^6 = 34012224.

$6^{18}$ is bigger: $6^{18}$ = 1.01559 × $10^{14}$ and $18^{6}$ = 34012224.

6^{18} = 1.01559\times 10^{14}\ >\ 18^{6} = 34012224

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^18 against 18^6 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: $6^{18}$ = 101559956668416 and $18^{6}$ = 34012224.
2
So $6^{18}$ 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 6 and 18 are past e, where the function is decreasing, so the larger exponent loses and $6^{18}$ > $18^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 18}{18}$

LaTeX source of the formulas on this page

\[ 6^{18} = 1.01559\times 10^{14}\ >\ 18^{6} = 34012224 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 18}{18} \]

Questions

Which is bigger,

6^{18}

18^{6}

$6^{18}$ is bigger: $6^{18}$ = 1.01559 × $10^{14}$ and $18^{6}$ = 34012224.

Which is bigger, 6186^{18}618 or 18618^{6}186?

How to work it out

Questions

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Which is bigger, $6^{18}$ or $18^{6}$ ?