Which is bigger, $3^{24}$ or $24^{3}$ ?

Q: Which is bigger, 3^24 or 24^3?

3^24 is bigger: 3^24 = 282429536481 and 24^3 = 13824.

$3^{24}$ is bigger: $3^{24}$ = 282429536481 and $24^{3}$ = 13824.

3^{24} = 282429536481\ >\ 24^{3} = 13824

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The curve ln x / x, which decides every comparison of a^b with b^a. 3^24 against 24^3 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: $3^{24}$ = 282429536481 and $24^{3}$ = 13824.
2
So $3^{24}$ 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 3 and 24 are past e, where the function is decreasing, so the larger exponent loses and $3^{24}$ > $24^{3}$ .
$\frac{\ln 3}{3}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 3^{24} = 282429536481\ >\ 24^{3} = 13824 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

3^{24}

24^{3}

$3^{24}$ is bigger: $3^{24}$ = 282429536481 and $24^{3}$ = 13824.

Which is bigger, 3243^{24}324 or 24324^{3}243?

How to work it out

Questions

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Which is bigger, $3^{24}$ or $24^{3}$ ?