Which is bigger, $8^{20}$ or $20^{8}$ ?

Q: Which is bigger, 8^20 or 20^8?

8^20 is bigger: 8^20 = 1.15292 × 10^18 and 20^8 = 25600000000.

$8^{20}$ is bigger: $8^{20}$ = 1.15292 × $10^{18}$ and $20^{8}$ = 25600000000.

8^{20} = 1.15292\times 10^{18}\ >\ 20^{8} = 25600000000

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The curve ln x / x, which decides every comparison of a^b with b^a. 8^20 against 20^8 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: $8^{20}$ = 1152921504606846976 and $20^{8}$ = 25600000000.
2
So $8^{20}$ 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 8 and 20 are past e, where the function is decreasing, so the larger exponent loses and $8^{20}$ > $20^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 8^{20} = 1.15292\times 10^{18}\ >\ 20^{8} = 25600000000 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

8^{20}

20^{8}

$8^{20}$ is bigger: $8^{20}$ = 1.15292 × $10^{18}$ and $20^{8}$ = 25600000000.

Which is bigger, 8208^{20}820 or 20820^{8}208?

How to work it out

Questions

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Which is bigger, $8^{20}$ or $20^{8}$ ?