Which is bigger, $12^{30}$ or $30^{12}$ ?

Q: Which is bigger, 12^30 or 30^12?

12^30 is bigger: 12^30 = 2.37376 × 10^32 and 30^12 = 5.31441 × 10^17.

$12^{30}$ is bigger: $12^{30}$ = 2.37376 × $10^{32}$ and $30^{12}$ = 5.31441 × $10^{17}$ .

12^{30} = 2.37376\times 10^{32}\ >\ 30^{12} = 5.31441\times 10^{17}

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The curve ln x / x, which decides every comparison of a^b with b^a. 12^30 against 30^12 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: $12^{30}$ = 237376313799769806328950291431424 and $30^{12}$ = 531441000000000000.
2
So $12^{30}$ 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 12 and 30 are past e, where the function is decreasing, so the larger exponent loses and $12^{30}$ > $30^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 30}{30}$

LaTeX source of the formulas on this page

\[ 12^{30} = 2.37376\times 10^{32}\ >\ 30^{12} = 5.31441\times 10^{17} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 30}{30} \]

Questions

Which is bigger,

12^{30}

30^{12}

$12^{30}$ is bigger: $12^{30}$ = 2.37376 × $10^{32}$ and $30^{12}$ = 5.31441 × $10^{17}$ .

Which is bigger, 123012^{30}1230 or 301230^{12}3012?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{30}$ or $30^{12}$ ?