Which is bigger, $6^{20}$ or $20^{6}$ ?

Q: Which is bigger, 6^20 or 20^6?

6^20 is bigger: 6^20 = 3.65615 × 10^15 and 20^6 = 64000000.

$6^{20}$ is bigger: $6^{20}$ = 3.65615 × $10^{15}$ and $20^{6}$ = 64000000.

6^{20} = 3.65615\times 10^{15}\ >\ 20^{6} = 64000000

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^20 against 20^6 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: $6^{20}$ = 3656158440062976 and $20^{6}$ = 64000000.
2
So $6^{20}$ 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 20 are past e, where the function is decreasing, so the larger exponent loses and $6^{20}$ > $20^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 6^{20} = 3.65615\times 10^{15}\ >\ 20^{6} = 64000000 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

6^{20}

20^{6}

$6^{20}$ is bigger: $6^{20}$ = 3.65615 × $10^{15}$ and $20^{6}$ = 64000000.

Which is bigger, 6206^{20}620 or 20620^{6}206?

How to work it out

Questions

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Which is bigger, $6^{20}$ or $20^{6}$ ?