Which is bigger, $17^{20}$ or $20^{17}$ ?

Q: Which is bigger, 17^20 or 20^17?

17^20 is bigger: 17^20 = 4.06423 × 10^24 and 20^17 = 1.31072 × 10^22.

$17^{20}$ is bigger: $17^{20}$ = 4.06423 × $10^{24}$ and $20^{17}$ = 1.31072 × $10^{22}$ .

17^{20} = 4.06423\times 10^{24}\ >\ 20^{17} = 1.31072\times 10^{22}

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. 17^20 against 20^17 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: $17^{20}$ = 4064231406647572522401601 and $20^{17}$ = 13107200000000000000000.
2
So $17^{20}$ 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 17 and 20 are past e, where the function is decreasing, so the larger exponent loses and $17^{20}$ > $20^{17}$ .
$\frac{\ln 17}{17}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 17^{20} = 4.06423\times 10^{24}\ >\ 20^{17} = 1.31072\times 10^{22} \]
\[ \frac{\ln 17}{17}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

17^{20}

20^{17}

$17^{20}$ is bigger: $17^{20}$ = 4.06423 × $10^{24}$ and $20^{17}$ = 1.31072 × $10^{22}$ .

Which is bigger, 172017^{20}1720 or 201720^{17}2017?

How to work it out

Questions

Similar comparisons

Which is bigger, $17^{20}$ or $20^{17}$ ?