Which is bigger, $7^{30}$ or $30^{7}$ ?

Q: Which is bigger, 7^30 or 30^7?

7^30 is bigger: 7^30 = 2.25393 × 10^25 and 30^7 = 21870000000.

$7^{30}$ is bigger: $7^{30}$ = 2.25393 × $10^{25}$ and $30^{7}$ = 21870000000.

7^{30} = 2.25393\times 10^{25}\ >\ 30^{7} = 21870000000

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The curve ln x / x, which decides every comparison of a^b with b^a. 7^30 against 30^7 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: $7^{30}$ = 22539340290692258087863249 and $30^{7}$ = 21870000000.
2
So $7^{30}$ 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 7 and 30 are past e, where the function is decreasing, so the larger exponent loses and $7^{30}$ > $30^{7}$ .
$\frac{\ln 7}{7}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 7^{30} = 2.25393\times 10^{25}\ >\ 30^{7} = 21870000000 \]
\[ \frac{\ln 7}{7}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

7^{30}

30^{7}

$7^{30}$ is bigger: $7^{30}$ = 2.25393 × $10^{25}$ and $30^{7}$ = 21870000000.

Which is bigger, 7307^{30}730 or 30730^{7}307?

How to work it out

Questions

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Which is bigger, $7^{30}$ or $30^{7}$ ?