Which is bigger, $4^{24}$ or $24^{4}$ ?

Q: Which is bigger, 4^24 or 24^4?

4^24 is bigger: 4^24 = 2.81474 × 10^14 and 24^4 = 331776.

$4^{24}$ is bigger: $4^{24}$ = 2.81474 × $10^{14}$ and $24^{4}$ = 331776.

4^{24} = 2.81474\times 10^{14}\ >\ 24^{4} = 331776

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^24 against 24^4 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: $4^{24}$ = 281474976710656 and $24^{4}$ = 331776.
2
So $4^{24}$ 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 24 are past e, where the function is decreasing, so the larger exponent loses and $4^{24}$ > $24^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 4^{24} = 2.81474\times 10^{14}\ >\ 24^{4} = 331776 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

4^{24}

24^{4}

$4^{24}$ is bigger: $4^{24}$ = 2.81474 × $10^{14}$ and $24^{4}$ = 331776.

Which is bigger, 4244^{24}424 or 24424^{4}244?

How to work it out

Questions

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Which is bigger, $4^{24}$ or $24^{4}$ ?