Evaluate log(216√6) to the base 6.

To evaluate \( \log_6 (216\sqrt{6}) \), we can break it down step by step using logarithmic properties.

We need to find the logarithm of \( 216\sqrt{6} \) with base 6.

First, express \( 216\sqrt{6} \) as a product of numbers:
\[ 216\sqrt{6} = 216 \times \sqrt{6}. \]

Now, let's express 216 and \( \sqrt{6} \) in terms of powers of 6.

We know:
- \( 216 = 6^3 \), because \( 6^3 = 6 \times 6 \times 6 = 216 \).
- \( \sqrt{6} = 6^{1/2} \), because \( 6^{1/2} \) is the square root of 6.

So, we have:
\[ 216\sqrt{6} = 6^3 \times 6^{1/2}. \]

Using the property of exponents, \( a^m \times a^n = a^{m+n} \), we can combine the terms:
\[ 6^3 \times 6^{1/2} = 6^{3 + 1/2} = 6^{7/2}. \]

Now, we can take the logarithm of both sides:
\[
\log_6 (216\sqrt{6}) = \log_6 (6^{7/2}).
\]

Using the property of logarithms, \( \log_b (a^n) = n \log_b (a) \), we get:
\[
\log_6 (6^{7/2}) = \frac{7}{2} \log_6 (6).
\]

Since \( \log_6 (6) = 1 \), the expression simplifies to:
\[
\frac{7}{2} \times 1 = \frac{7}{2}.
\]

Thus, the value of \( \log_6 (216\sqrt{6}) \) is \( \frac{7}{2} \).