Which is bigger, $6^{8}$ or $8^{6}$ ?

Q: Which is bigger, 6^8 or 8^6?

6^8 is bigger: 6^8 = 1679616 and 8^6 = 262144.

$6^{8}$ is bigger: $6^{8}$ = 1679616 and $8^{6}$ = 262144.

6^{8} = 1679616\ >\ 8^{6} = 262144

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^8 against 8^6 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: $6^{8}$ = 1679616 and $8^{6}$ = 262144.
2
So $6^{8}$ 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 6 and 8 are past e, where the function is decreasing, so the larger exponent loses and $6^{8}$ > $8^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 8}{8}$

LaTeX source of the formulas on this page

\[ 6^{8} = 1679616\ >\ 8^{6} = 262144 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 8}{8} \]

Questions

Which is bigger,

6^{8}

8^{6}

$6^{8}$ is bigger: $6^{8}$ = 1679616 and $8^{6}$ = 262144.

Which is bigger, 686^{8}68 or 868^{6}86?

How to work it out

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Which is bigger, $6^{8}$ or $8^{6}$ ?