Which is bigger, $14^{16}$ or $16^{14}$ ?

Q: Which is bigger, 14^16 or 16^14?

14^16 is bigger: 14^16 = 2.17795 × 10^18 and 16^14 = 7.20575 × 10^16.

$14^{16}$ is bigger: $14^{16}$ = 2.17795 × $10^{18}$ and $16^{14}$ = 7.20575 × $10^{16}$ .

14^{16} = 2.17795\times 10^{18}\ >\ 16^{14} = 7.20575\times 10^{16}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^16 against 16^14 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: $14^{16}$ = 2177953337809371136 and $16^{14}$ = 72057594037927936.
2
So $14^{16}$ 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 16 are past e, where the function is decreasing, so the larger exponent loses and $14^{16}$ > $16^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 16}{16}$

LaTeX source of the formulas on this page

\[ 14^{16} = 2.17795\times 10^{18}\ >\ 16^{14} = 7.20575\times 10^{16} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 16}{16} \]

Questions

Which is bigger,

14^{16}

16^{14}

$14^{16}$ is bigger: $14^{16}$ = 2.17795 × $10^{18}$ and $16^{14}$ = 7.20575 × $10^{16}$ .

Which is bigger, 141614^{16}1416 or 161416^{14}1614?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{16}$ or $16^{14}$ ?