Which is bigger, $12^{20}$ or $20^{12}$ ?

Q: Which is bigger, 12^20 or 20^12?

12^20 is bigger: 12^20 = 3.83375 × 10^21 and 20^12 = 4.09600 × 10^15.

$12^{20}$ is bigger: $12^{20}$ = 3.83375 × $10^{21}$ and $20^{12}$ = 4.09600 × $10^{15}$ .

12^{20} = 3.83375\times 10^{21}\ >\ 20^{12} = 4.09600\times 10^{15}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^20 against 20^12 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: $12^{20}$ = 3833759992447475122176 and $20^{12}$ = 4096000000000000.
2
So $12^{20}$ 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 20 are past e, where the function is decreasing, so the larger exponent loses and $12^{20}$ > $20^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 12^{20} = 3.83375\times 10^{21}\ >\ 20^{12} = 4.09600\times 10^{15} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

12^{20}

20^{12}

$12^{20}$ is bigger: $12^{20}$ = 3.83375 × $10^{21}$ and $20^{12}$ = 4.09600 × $10^{15}$ .

Which is bigger, 122012^{20}1220 or 201220^{12}2012?

How to work it out

Questions

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Which is bigger, $12^{20}$ or $20^{12}$ ?