Which is bigger, $14^{28}$ or $28^{14}$ ?

Q: Which is bigger, 14^28 or 28^14?

14^28 is bigger: 14^28 = 1.23476 × 10^32 and 28^14 = 1.82059 × 10^20.

$14^{28}$ is bigger: $14^{28}$ = 1.23476 × $10^{32}$ and $28^{14}$ = 1.82059 × $10^{20}$ .

14^{28} = 1.23476\times 10^{32}\ >\ 28^{14} = 1.82059\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. 14^28 against 28^14 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: $14^{28}$ = 123476695691247935826229781856256 and $28^{14}$ = 182059119829942534144.
2
So $14^{28}$ 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 28 are past e, where the function is decreasing, so the larger exponent loses and $14^{28}$ > $28^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 28}{28}$

LaTeX source of the formulas on this page

\[ 14^{28} = 1.23476\times 10^{32}\ >\ 28^{14} = 1.82059\times 10^{20} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 28}{28} \]

Questions

Which is bigger,

14^{28}

28^{14}

$14^{28}$ is bigger: $14^{28}$ = 1.23476 × $10^{32}$ and $28^{14}$ = 1.82059 × $10^{20}$ .

Which is bigger, 142814^{28}1428 or 281428^{14}2814?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{28}$ or $28^{14}$ ?