Which is bigger, $6^{21}$ or $21^{6}$ ?

Q: Which is bigger, 6^21 or 21^6?

6^21 is bigger: 6^21 = 2.19369 × 10^16 and 21^6 = 85766121.

$6^{21}$ is bigger: $6^{21}$ = 2.19369 × $10^{16}$ and $21^{6}$ = 85766121.

6^{21} = 2.19369\times 10^{16}\ >\ 21^{6} = 85766121

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^21 against 21^6 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: $6^{21}$ = 21936950640377856 and $21^{6}$ = 85766121.
2
So $6^{21}$ 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 6 and 21 are past e, where the function is decreasing, so the larger exponent loses and $6^{21}$ > $21^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 21}{21}$

LaTeX source of the formulas on this page

\[ 6^{21} = 2.19369\times 10^{16}\ >\ 21^{6} = 85766121 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger,

6^{21}

21^{6}

$6^{21}$ is bigger: $6^{21}$ = 2.19369 × $10^{16}$ and $21^{6}$ = 85766121.

Which is bigger, 6216^{21}621 or 21621^{6}216?

How to work it out

Questions

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Which is bigger, $6^{21}$ or $21^{6}$ ?