Which is bigger, $16^{20}$ or $20^{16}$ ?

Q: Which is bigger, 16^20 or 20^16?

16^20 is bigger: 16^20 = 1.20892 × 10^24 and 20^16 = 6.55360 × 10^20.

$16^{20}$ is bigger: $16^{20}$ = 1.20892 × $10^{24}$ and $20^{16}$ = 6.55360 × $10^{20}$ .

16^{20} = 1.20892\times 10^{24}\ >\ 20^{16} = 6.55360\times 10^{20}

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. 16^20 against 20^16 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: $16^{20}$ = 1208925819614629174706176 and $20^{16}$ = 655360000000000000000.
2
So $16^{20}$ 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 16 and 20 are past e, where the function is decreasing, so the larger exponent loses and $16^{20}$ > $20^{16}$ .
$\frac{\ln 16}{16}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 16^{20} = 1.20892\times 10^{24}\ >\ 20^{16} = 6.55360\times 10^{20} \]
\[ \frac{\ln 16}{16}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

16^{20}

20^{16}

$16^{20}$ is bigger: $16^{20}$ = 1.20892 × $10^{24}$ and $20^{16}$ = 6.55360 × $10^{20}$ .

Which is bigger, 162016^{20}1620 or 201620^{16}2016?

How to work it out

Questions

Similar comparisons

Which is bigger, $16^{20}$ or $20^{16}$ ?