Which is bigger, $7^{14}$ or $14^{7}$ ?

Q: Which is bigger, 7^14 or 14^7?

7^14 is bigger: 7^14 = 678223072849 and 14^7 = 105413504.

$7^{14}$ is bigger: $7^{14}$ = 678223072849 and $14^{7}$ = 105413504.

7^{14} = 678223072849\ >\ 14^{7} = 105413504

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^14 against 14^7 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: $7^{14}$ = 678223072849 and $14^{7}$ = 105413504.
2
So $7^{14}$ 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 7 and 14 are past e, where the function is decreasing, so the larger exponent loses and $7^{14}$ > $14^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 14}{14}$

LaTeX source of the formulas on this page

\[ 7^{14} = 678223072849\ >\ 14^{7} = 105413504 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 14}{14} \]

Questions

Which is bigger,

7^{14}

14^{7}

$7^{14}$ is bigger: $7^{14}$ = 678223072849 and $14^{7}$ = 105413504.

Which is bigger, 7147^{14}714 or 14714^{7}147?

How to work it out

Questions

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Which is bigger, $7^{14}$ or $14^{7}$ ?