Which is bigger, $10^{24}$ or $24^{10}$ ?

Q: Which is bigger, 10^24 or 24^10?

10^24 is bigger: 10^24 = 1.00000 × 10^24 and 24^10 = 6.34033 × 10^13.

$10^{24}$ is bigger: $10^{24}$ = 1.00000 × $10^{24}$ and $24^{10}$ = 6.34033 × $10^{13}$ .

10^{24} = 1.00000\times 10^{24}\ >\ 24^{10} = 63403380965376

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The curve ln x / x, which decides every comparison of a^b with b^a. 10^24 against 24^10 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: $10^{24}$ = 1000000000000000000000000 and $24^{10}$ = 63403380965376.
2
So $10^{24}$ 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 10 and 24 are past e, where the function is decreasing, so the larger exponent loses and $10^{24}$ > $24^{10}$ .
$\frac{\ln 10}{10}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 10^{24} = 1.00000\times 10^{24}\ >\ 24^{10} = 63403380965376 \]
\[ \frac{\ln 10}{10}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

10^{24}

24^{10}

$10^{24}$ is bigger: $10^{24}$ = 1.00000 × $10^{24}$ and $24^{10}$ = 6.34033 × $10^{13}$ .

Which is bigger, 102410^{24}1024 or 241024^{10}2410?

How to work it out

Questions

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Which is bigger, $10^{24}$ or $24^{10}$ ?