Which is bigger, $24^{30}$ or $30^{24}$ ?

Q: Which is bigger, 24^30 or 30^24?

24^30 is bigger: 24^30 = 2.54880 × 10^41 and 30^24 = 2.82429 × 10^35.

$24^{30}$ is bigger: $24^{30}$ = 2.54880 × $10^{41}$ and $30^{24}$ = 2.82429 × $10^{35}$ .

24^{30} = 2.54880\times 10^{41}\ >\ 30^{24} = 2.82429\times 10^{35}

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The curve ln x / x, which decides every comparison of a^b with b^a. 24^30 against 30^24 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: $24^{30}$ = 254880876153761202627773829926908776677376 and $30^{24}$ = 282429536481000000000000000000000000.
2
So $24^{30}$ 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 24 and 30 are past e, where the function is decreasing, so the larger exponent loses and $24^{30}$ > $30^{24}$ .
$\frac{\ln 24}{24}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 24^{30} = 2.54880\times 10^{41}\ >\ 30^{24} = 2.82429\times 10^{35} \]
\[ \frac{\ln 24}{24}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

24^{30}

30^{24}

$24^{30}$ is bigger: $24^{30}$ = 2.54880 × $10^{41}$ and $30^{24}$ = 2.82429 × $10^{35}$ .

Which is bigger, 243024^{30}2430 or 302430^{24}3024?

How to work it out

Questions

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Which is bigger, $24^{30}$ or $30^{24}$ ?