Which is bigger, $8^{22}$ or $22^{8}$ ?

Q: Which is bigger, 8^22 or 22^8?

8^22 is bigger: 8^22 = 7.37869 × 10^19 and 22^8 = 54875873536.

$8^{22}$ is bigger: $8^{22}$ = 7.37869 × $10^{19}$ and $22^{8}$ = 54875873536.

8^{22} = 7.37869\times 10^{19}\ >\ 22^{8} = 54875873536

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The curve ln x / x, which decides every comparison of a^b with b^a. 8^22 against 22^8 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: $8^{22}$ = 73786976294838206464 and $22^{8}$ = 54875873536.
2
So $8^{22}$ 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 22 are past e, where the function is decreasing, so the larger exponent loses and $8^{22}$ > $22^{8}$ .
$\frac{\ln 8}{8}\ >\ \frac{\ln 22}{22}$

LaTeX source of the formulas on this page

\[ 8^{22} = 7.37869\times 10^{19}\ >\ 22^{8} = 54875873536 \]
\[ \frac{\ln 8}{8}\ >\ \frac{\ln 22}{22} \]

Questions

Which is bigger,

8^{22}

22^{8}

$8^{22}$ is bigger: $8^{22}$ = 7.37869 × $10^{19}$ and $22^{8}$ = 54875873536.

Which is bigger, 8228^{22}822 or 22822^{8}228?

How to work it out

Questions

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Which is bigger, $8^{22}$ or $22^{8}$ ?