Which is bigger, $4^{30}$ or $30^{4}$ ?

Q: Which is bigger, 4^30 or 30^4?

4^30 is bigger: 4^30 = 1.15292 × 10^18 and 30^4 = 810000.

$4^{30}$ is bigger: $4^{30}$ = 1.15292 × $10^{18}$ and $30^{4}$ = 810000.

4^{30} = 1.15292\times 10^{18}\ >\ 30^{4} = 810000

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^30 against 30^4 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: $4^{30}$ = 1152921504606846976 and $30^{4}$ = 810000.
2
So $4^{30}$ 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 4 and 30 are past e, where the function is decreasing, so the larger exponent loses and $4^{30}$ > $30^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 4^{30} = 1.15292\times 10^{18}\ >\ 30^{4} = 810000 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

4^{30}

30^{4}

$4^{30}$ is bigger: $4^{30}$ = 1.15292 × $10^{18}$ and $30^{4}$ = 810000.

Which is bigger, 4304^{30}430 or 30430^{4}304?

How to work it out

Questions

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Which is bigger, $4^{30}$ or $30^{4}$ ?