Which is bigger, $12^{16}$ or $16^{12}$ ?

Q: Which is bigger, 12^16 or 16^12?

12^16 is bigger: 12^16 = 1.84884 × 10^17 and 16^12 = 2.81474 × 10^14.

$12^{16}$ is bigger: $12^{16}$ = 1.84884 × $10^{17}$ and $16^{12}$ = 2.81474 × $10^{14}$ .

12^{16} = 1.84884\times 10^{17}\ >\ 16^{12} = 2.81474\times 10^{14}

Explain it step by step in chat Make a short video of it

The curve ln x / x, which decides every comparison of a^b with b^a. 12^16 against 16^12 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: $12^{16}$ = 184884258895036416 and $16^{12}$ = 281474976710656.
2
So $12^{16}$ 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 16 are past e, where the function is decreasing, so the larger exponent loses and $12^{16}$ > $16^{12}$ .
$\frac{\ln 12}{12}\ >\ \frac{\ln 16}{16}$

LaTeX source of the formulas on this page

\[ 12^{16} = 1.84884\times 10^{17}\ >\ 16^{12} = 2.81474\times 10^{14} \]
\[ \frac{\ln 12}{12}\ >\ \frac{\ln 16}{16} \]

Questions

Which is bigger,

12^{16}

16^{12}

$12^{16}$ is bigger: $12^{16}$ = 1.84884 × $10^{17}$ and $16^{12}$ = 2.81474 × $10^{14}$ .

Which is bigger, 121612^{16}1216 or 161216^{12}1612?

How to work it out

Questions

Similar comparisons

Which is bigger, $12^{16}$ or $16^{12}$ ?