Which is bigger, $4^{22}$ or $22^{4}$ ?

Q: Which is bigger, 4^22 or 22^4?

4^22 is bigger: 4^22 = 1.75921 × 10^13 and 22^4 = 234256.

$4^{22}$ is bigger: $4^{22}$ = 1.75921 × $10^{13}$ and $22^{4}$ = 234256.

4^{22} = 17592186044416\ >\ 22^{4} = 234256

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^22 against 22^4 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: $4^{22}$ = 17592186044416 and $22^{4}$ = 234256.
2
So $4^{22}$ 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 22 are past e, where the function is decreasing, so the larger exponent loses and $4^{22}$ > $22^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 4^{22} = 17592186044416\ >\ 22^{4} = 234256 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

4^{22}

22^{4}

$4^{22}$ is bigger: $4^{22}$ = 1.75921 × $10^{13}$ and $22^{4}$ = 234256.

Which is bigger, 4224^{22}422 or 22422^{4}224?

How to work it out

Questions

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Which is bigger, $4^{22}$ or $22^{4}$ ?