Which is bigger, $4^{8}$ or $8^{4}$ ?

Q: Which is bigger, 4^8 or 8^4?

4^8 is bigger: 4^8 = 65536 and 8^4 = 4096.

$4^{8}$ is bigger: $4^{8}$ = 65536 and $8^{4}$ = 4096.

4^{8} = 65536\ >\ 8^{4} = 4096

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The curve ln x / x, which decides every comparison of a^b with b^a. 4^8 against 8^4 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: $4^{8}$ = 65536 and $8^{4}$ = 4096.
2
So $4^{8}$ 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 4 and 8 are past e, where the function is decreasing, so the larger exponent loses and $4^{8}$ > $8^{4}$ .
$\frac{\ln 4}{4}\ >\ \frac{\ln 8}{8}$

LaTeX source of the formulas on this page

\[ 4^{8} = 65536\ >\ 8^{4} = 4096 \]
\[ \frac{\ln 4}{4}\ >\ \frac{\ln 8}{8} \]

Questions

Which is bigger,

4^{8}

8^{4}

$4^{8}$ is bigger: $4^{8}$ = 65536 and $8^{4}$ = 4096.

Which is bigger, 484^{8}48 or 848^{4}84?

How to work it out

Questions

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Which is bigger, $4^{8}$ or $8^{4}$ ?