Which is bigger, $14^{21}$ or $21^{14}$ ?

Q: Which is bigger, 14^21 or 21^14?

14^21 is bigger: 14^21 = 1.17135 × 10^24 and 21^14 = 3.24391 × 10^18.

$14^{21}$ is bigger: $14^{21}$ = 1.17135 × $10^{24}$ and $21^{14}$ = 3.24391 × $10^{18}$ .

14^{21} = 1.17135\times 10^{24}\ >\ 21^{14} = 3.24391\times 10^{18}

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The curve ln x / x, which decides every comparison of a^b with b^a. 14^21 against 21^14 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: $14^{21}$ = 1171355575953987221848064 and $21^{14}$ = 3243919932521508681.
2
So $14^{21}$ 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 21 are past e, where the function is decreasing, so the larger exponent loses and $14^{21}$ > $21^{14}$ .
$\frac{\ln 14}{14}\ >\ \frac{\ln 21}{21}$

LaTeX source of the formulas on this page

\[ 14^{21} = 1.17135\times 10^{24}\ >\ 21^{14} = 3.24391\times 10^{18} \]
\[ \frac{\ln 14}{14}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger,

14^{21}

21^{14}

$14^{21}$ is bigger: $14^{21}$ = 1.17135 × $10^{24}$ and $21^{14}$ = 3.24391 × $10^{18}$ .

Which is bigger, 142114^{21}1421 or 211421^{14}2114?

How to work it out

Questions

Similar comparisons

Which is bigger, $14^{21}$ or $21^{14}$ ?