Which is bigger, $12^{24}$ or $24^{12}$ ?

Q: Which is bigger, 12^24 or 24^12?

12^24 is bigger: 12^24 = 7.94968 × 10^25 and 24^12 = 3.65203 × 10^16.

$12^{24}$ is bigger: $12^{24}$ = 7.94968 × $10^{25}$ and $24^{12}$ = 3.65203 × $10^{16}$ .

12^{24} = 7.94968\times 10^{25}\ >\ 24^{12} = 3.65203\times 10^{16}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^24 against 24^12 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: $12^{24}$ = 79496847203390844133441536 and $24^{12}$ = 36520347436056576.
2
So $12^{24}$ 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 12 and 24 are past e, where the function is decreasing, so the larger exponent loses and $12^{24}$ > $24^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 12^{24} = 7.94968\times 10^{25}\ >\ 24^{12} = 3.65203\times 10^{16} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

12^{24}

24^{12}

$12^{24}$ is bigger: $12^{24}$ = 7.94968 × $10^{25}$ and $24^{12}$ = 3.65203 × $10^{16}$ .

Which is bigger, 122412^{24}1224 or 241224^{12}2412?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{24}$ or $24^{12}$ ?