`log216sqrt(6)` to the base 6 is equal to

Options

A. A)3
B. B)`3//2`
C. C)`7//2`
D. D)None of these

Answer

To solve the problem \( \log_{6}(216\sqrt{6}) \), we will break it down step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ \log_{6}(216\sqrt{6}) \]
We can rewrite \( 216\sqrt{6} \) as \( 216 \cdot 6^{1/2} \).

### Step 2: Use the property of logarithms
Using the property of logarithms that states \( \log_b(m \cdot n) = \log_b(m) + \log_b(n) \), we can split the logarithm:
\[ \log_{6}(216\sqrt{6}) = \log_{6}(216) + \log_{6}(\sqrt{6}) \]

### Step 3: Simplify \( \log_{6}(\sqrt{6}) \)
The term \( \sqrt{6} \) can be rewritten as \( 6^{1/2} \). Applying the logarithmic property \( \log_b(a^c) = c \cdot \log_b(a) \):
\[ \log_{6}(\sqrt{6}) = \log_{6}(6^{1/2}) = \frac{1}{2} \cdot \log_{6}(6) \]
Since \( \log_{6}(6) = 1 \):
\[ \log_{6}(\sqrt{6}) = \frac{1}{2} \]

### Step 4: Calculate \( \log_{6}(216) \)
Next, we need to express \( 216 \) in terms of base \( 6 \). We know that:
\[ 216 = 6^3 \]
Thus:
\[ \log_{6}(216) = \log_{6}(6^3) = 3 \cdot \log_{6}(6) = 3 \cdot 1 = 3 \]

### Step 5: Combine the results
Now we can combine our results:
\[ \log_{6}(216\sqrt{6}) = \log_{6}(216) + \log_{6}(\sqrt{6}) = 3 + \frac{1}{2} \]
To add these, convert \( 3 \) into a fraction:
\[ 3 = \frac{6}{2} \]
So:
\[ \log_{6}(216\sqrt{6}) = \frac{6}{2} + \frac{1}{2} = \frac{7}{2} \]

### Final Answer
Thus, the final answer is:
\[ \log_{6}(216\sqrt{6}) = \frac{7}{2} \]