Which is bigger, $14^{22}$ or $22^{14}$ ?

Q: Which is bigger, 14^22 or 22^14?

14^22 is bigger: 14^22 = 1.63989 × 10^25 and 22^14 = 6.22182 × 10^18.

$14^{22}$ is bigger: $14^{22}$ = 1.63989 × $10^{25}$ and $22^{14}$ = 6.22182 × $10^{18}$ .

14^{22} = 1.63989\times 10^{25}\ >\ 22^{14} = 6.22182\times 10^{18}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^22 against 22^14 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: $14^{22}$ = 16398978063355821105872896 and $22^{14}$ = 6221821273427820544.
2
So $14^{22}$ 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 22 are past e, where the function is decreasing, so the larger exponent loses and $14^{22}$ > $22^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 14^{22} = 1.63989\times 10^{25}\ >\ 22^{14} = 6.22182\times 10^{18} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

14^{22}

22^{14}

$14^{22}$ is bigger: $14^{22}$ = 1.63989 × $10^{25}$ and $22^{14}$ = 6.22182 × $10^{18}$ .

Which is bigger, 142214^{22}1422 or 221422^{14}2214?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{22}$ or $22^{14}$ ?