Which is bigger, $27^{30}$ or $30^{27}$ ?

Q: Which is bigger, 27^30 or 30^27?

27^30 is bigger: 27^30 = 8.72796 × 10^42 and 30^27 = 7.62559 × 10^39.

$27^{30}$ is bigger: $27^{30}$ = 8.72796 × $10^{42}$ and $30^{27}$ = 7.62559 × $10^{39}$ .

27^{30} = 8.72796\times 10^{42}\ >\ 30^{27} = 7.62559\times 10^{39}

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The curve ln x / x, which decides every comparison of a^b with b^a. 27^30 against 30^27 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: $27^{30}$ = 8727963568087712425891397479476727340041449 and $30^{27}$ = 7625597484987000000000000000000000000000.
2
So $27^{30}$ 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 27 and 30 are past e, where the function is decreasing, so the larger exponent loses and $27^{30}$ > $30^{27}$ .
$\frac{\ln 27}{27}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 27^{30} = 8.72796\times 10^{42}\ >\ 30^{27} = 7.62559\times 10^{39} \]
\[ \frac{\ln 27}{27}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

27^{30}

30^{27}

$27^{30}$ is bigger: $27^{30}$ = 8.72796 × $10^{42}$ and $30^{27}$ = 7.62559 × $10^{39}$ .

Which is bigger, 273027^{30}2730 or 302730^{27}3027?

How to work it out

Questions

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Which is bigger, $27^{30}$ or $30^{27}$ ?