Which is bigger, $14^{27}$ or $27^{14}$ ?

Q: Which is bigger, 14^27 or 27^14?

14^27 is bigger: 14^27 = 8.81976 × 10^30 and 27^14 = 1.09418 × 10^20.

$14^{27}$ is bigger: $14^{27}$ = 8.81976 × $10^{30}$ and $27^{14}$ = 1.09418 × $10^{20}$ .

14^{27} = 8.81976\times 10^{30}\ >\ 27^{14} = 1.09418\times 10^{20}

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. 14^27 against 27^14 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: $14^{27}$ = 8819763977946281130444984418304 and $27^{14}$ = 109418989131512359209.
2
So $14^{27}$ 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 14 and 27 are past e, where the function is decreasing, so the larger exponent loses and $14^{27}$ > $27^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 27}{27}$

LaTeX source of the formulas on this page

\[ 14^{27} = 8.81976\times 10^{30}\ >\ 27^{14} = 1.09418\times 10^{20} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 27}{27} \]

Questions

Which is bigger,

14^{27}

27^{14}

$14^{27}$ is bigger: $14^{27}$ = 8.81976 × $10^{30}$ and $27^{14}$ = 1.09418 × $10^{20}$ .

Which is bigger, 142714^{27}1427 or 271427^{14}2714?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{27}$ or $27^{14}$ ?