Which is bigger, $6^{11}$ or $11^{6}$ ?

Q: Which is bigger, 6^11 or 11^6?

6^11 is bigger: 6^11 = 362797056 and 11^6 = 1771561.

$6^{11}$ is bigger: $6^{11}$ = 362797056 and $11^{6}$ = 1771561.

6^{11} = 362797056\ >\ 11^{6} = 1771561

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The curve ln x / x, which decides every comparison of a^b with b^a. 6^11 against 11^6 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: $6^{11}$ = 362797056 and $11^{6}$ = 1771561.
2
So $6^{11}$ 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 11 are past e, where the function is decreasing, so the larger exponent loses and $6^{11}$ > $11^{6}$ .
$\frac{\ln 6}{6}\ >\ \frac{\ln 11}{11}$

LaTeX source of the formulas on this page

\[ 6^{11} = 362797056\ >\ 11^{6} = 1771561 \]
\[ \frac{\ln 6}{6}\ >\ \frac{\ln 11}{11} \]

Questions

Which is bigger,

6^{11}

11^{6}

$6^{11}$ is bigger: $6^{11}$ = 362797056 and $11^{6}$ = 1771561.

Which is bigger, 6116^{11}611 or 11611^{6}116?

How to work it out

Questions

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Which is bigger, $6^{11}$ or $11^{6}$ ?