Which is bigger, $11^{14}$ or $14^{11}$ ?

Q: Which is bigger, 11^14 or 14^11?

11^14 is bigger: 11^14 = 3.79749 × 10^14 and 14^11 = 4.04956 × 10^12.

$11^{14}$ is bigger: $11^{14}$ = 3.79749 × $10^{14}$ and $14^{11}$ = 4.04956 × $10^{12}$ .

11^{14} = 3.79749\times 10^{14}\ >\ 14^{11} = 4049565169664

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. 11^14 against 14^11 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: $11^{14}$ = 379749833583241 and $14^{11}$ = 4049565169664.
2
So $11^{14}$ 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 11 and 14 are past e, where the function is decreasing, so the larger exponent loses and $11^{14}$ > $14^{11}$ .
$\frac{\ln 11}{11}\ >\ \frac{\ln 14}{14}$

LaTeX source of the formulas on this page

\[ 11^{14} = 3.79749\times 10^{14}\ >\ 14^{11} = 4049565169664 \]
\[ \frac{\ln 11}{11}\ >\ \frac{\ln 14}{14} \]

Questions

Which is bigger,

11^{14}

14^{11}

$11^{14}$ is bigger: $11^{14}$ = 3.79749 × $10^{14}$ and $14^{11}$ = 4.04956 × $10^{12}$ .

Which is bigger, 111411^{14}1114 or 141114^{11}1411?

How to work it out

Questions

Similar comparisons

Which is bigger, $11^{14}$ or $14^{11}$ ?