Which is bigger, $13^{14}$ or $14^{13}$ ?

Q: Which is bigger, 13^14 or 14^13?

13^14 is bigger: 13^14 = 3.93737 × 10^15 and 14^13 = 7.93714 × 10^14.

$13^{14}$ is bigger: $13^{14}$ = 3.93737 × $10^{15}$ and $14^{13}$ = 7.93714 × $10^{14}$ .

13^{14} = 3.93737\times 10^{15}\ >\ 14^{13} = 7.93714\times 10^{14}

Explain it step by step in chat Make a short video of it

The curve ln x / x, which decides every comparison of a^b with b^a. 13^14 against 14^13 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: $13^{14}$ = 3937376385699289 and $14^{13}$ = 793714773254144.
2
So $13^{14}$ 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 13 and 14 are past e, where the function is decreasing, so the larger exponent loses and $13^{14}$ > $14^{13}$ .
$\frac{\ln 13}{13}\ >\ \frac{\ln 14}{14}$

LaTeX source of the formulas on this page

\[ 13^{14} = 3.93737\times 10^{15}\ >\ 14^{13} = 7.93714\times 10^{14} \]
\[ \frac{\ln 13}{13}\ >\ \frac{\ln 14}{14} \]

Questions

Which is bigger,

13^{14}

14^{13}

$13^{14}$ is bigger: $13^{14}$ = 3.93737 × $10^{15}$ and $14^{13}$ = 7.93714 × $10^{14}$ .

Which is bigger, 131413^{14}1314 or 141314^{13}1413?

How to work it out

Questions

Similar comparisons

Which is bigger, $13^{14}$ or $14^{13}$ ?