The value of `log_sqrt(6)216` is :

Options

A. `1/6`
B. `sqrt6`
C. 3
D. 6

Answer

To find the value of \( \log_{\sqrt{6}} 216 \), we can follow these steps:

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

### Step 2: Use the change of base formula
Using the property of logarithms that states \( \log_{a^b} c = \frac{1}{b} \log_a c \), we can rewrite our expression:
\[ \log_{6^{1/2}} 216 = \frac{1}{1/2} \log_6 216 = 2 \log_6 216 \]

### Step 3: Factor 216
Next, we need to express 216 in terms of base 6. We know that:
\[ 216 = 6^3 \]
Thus, we can substitute this into our logarithm:
\[ \log_6 216 = \log_6 (6^3) \]

### Step 4: Apply the power rule of logarithms
Using the property of logarithms that states \( \log_a (a^b) = b \), we can simplify:
\[ \log_6 (6^3) = 3 \]

### Step 5: Substitute back into the expression
Now we substitute back into our expression for \( \log_{\sqrt{6}} 216 \):
\[ 2 \log_6 216 = 2 \times 3 = 6 \]

### Final Answer
Thus, the value of \( \log_{\sqrt{6}} 216 \) is:
\[ \boxed{6} \]