Which is bigger, $3^{20}$ or $20^{3}$ ?

Q: Which is bigger, 3^20 or 20^3?

3^20 is bigger: 3^20 = 3486784401 and 20^3 = 8000.

$3^{20}$ is bigger: $3^{20}$ = 3486784401 and $20^{3}$ = 8000.

3^{20} = 3486784401\ >\ 20^{3} = 8000

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The curve ln x / x, which decides every comparison of a^b with b^a. 3^20 against 20^3 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: $3^{20}$ = 3486784401 and $20^{3}$ = 8000.
2
So $3^{20}$ 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 3 and 20 are past e, where the function is decreasing, so the larger exponent loses and $3^{20}$ > $20^{3}$ .
$\frac{\ln 3}{3}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 3^{20} = 3486784401\ >\ 20^{3} = 8000 \]
\[ \frac{\ln 3}{3}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

3^{20}

20^{3}

$3^{20}$ is bigger: $3^{20}$ = 3486784401 and $20^{3}$ = 8000.

Which is bigger, 3203^{20}320 or 20320^{3}203?

How to work it out

Questions

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Which is bigger, $3^{20}$ or $20^{3}$ ?