Which is bigger, $6^{30}$ or $30^{6}$ ?

Q: Which is bigger, 6^30 or 30^6?

6^30 is bigger: 6^30 = 2.21073 × 10^23 and 30^6 = 729000000.

$6^{30}$ is bigger: $6^{30}$ = 2.21073 × $10^{23}$ and $30^{6}$ = 729000000.

6^{30} = 2.21073\times 10^{23}\ >\ 30^{6} = 729000000

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^30 against 30^6 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: $6^{30}$ = 221073919720733357899776 and $30^{6}$ = 729000000.
2
So $6^{30}$ 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 30 are past e, where the function is decreasing, so the larger exponent loses and $6^{30}$ > $30^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 6^{30} = 2.21073\times 10^{23}\ >\ 30^{6} = 729000000 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

6^{30}

30^{6}

$6^{30}$ is bigger: $6^{30}$ = 2.21073 × $10^{23}$ and $30^{6}$ = 729000000.

Which is bigger, 6306^{30}630 or 30630^{6}306?

How to work it out

Questions

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Which is bigger, $6^{30}$ or $30^{6}$ ?