Which is bigger, $14^{18}$ or $18^{14}$ ?

Q: Which is bigger, 14^18 or 18^14?

14^18 is bigger: 14^18 = 4.26878 × 10^20 and 18^14 = 3.74813 × 10^17.

$14^{18}$ is bigger: $14^{18}$ = 4.26878 × $10^{20}$ and $18^{14}$ = 3.74813 × $10^{17}$ .

14^{18} = 4.26878\times 10^{20}\ >\ 18^{14} = 3.74813\times 10^{17}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^18 against 18^14 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: $14^{18}$ = 426878854210636742656 and $18^{14}$ = 374813367582081024.
2
So $14^{18}$ 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 18 are past e, where the function is decreasing, so the larger exponent loses and $14^{18}$ > $18^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 18}{18}$

LaTeX source of the formulas on this page

\[ 14^{18} = 4.26878\times 10^{20}\ >\ 18^{14} = 3.74813\times 10^{17} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 18}{18} \]

Questions

Which is bigger,

14^{18}

18^{14}

$14^{18}$ is bigger: $14^{18}$ = 4.26878 × $10^{20}$ and $18^{14}$ = 3.74813 × $10^{17}$ .

Which is bigger, 141814^{18}1418 or 181418^{14}1814?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{18}$ or $18^{14}$ ?