Which is bigger, $6^{14}$ or $14^{6}$ ?

Q: Which is bigger, 6^14 or 14^6?

6^14 is bigger: 6^14 = 78364164096 and 14^6 = 7529536.

$6^{14}$ is bigger: $6^{14}$ = 78364164096 and $14^{6}$ = 7529536.

6^{14} = 78364164096\ >\ 14^{6} = 7529536

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^14 against 14^6 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: $6^{14}$ = 78364164096 and $14^{6}$ = 7529536.
2
So $6^{14}$ 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 6 and 14 are past e, where the function is decreasing, so the larger exponent loses and $6^{14}$ > $14^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 14}{14}$

LaTeX source of the formulas on this page

\[ 6^{14} = 78364164096\ >\ 14^{6} = 7529536 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 14}{14} \]

Questions

Which is bigger,

6^{14}

14^{6}

$6^{14}$ is bigger: $6^{14}$ = 78364164096 and $14^{6}$ = 7529536.

Which is bigger, 6146^{14}614 or 14614^{6}146?

How to work it out

Questions

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Which is bigger, $6^{14}$ or $14^{6}$ ?