Find the value of `(8log_(8)8)/(2log_(sqrt(8))8)`

Options

A. A)1
B. B)2
C. C)3
D. D)4

Answer

To solve the expression \((8 \log_{8} 8) / (2 \log_{\sqrt{8}} 8)\), we will follow these steps:

### Step 1: Simplify \( \log_{8} 8 \)
Using the property of logarithms that states \(\log_{a} a = 1\), we can simplify:
\[ \log_{8} 8 = 1 \]

### Step 2: Substitute into the expression
Now, substitute \(\log_{8} 8\) back into the expression:
\[ \frac{8 \cdot 1}{2 \log_{\sqrt{8}} 8} = \frac{8}{2 \log_{\sqrt{8}} 8} \]

### Step 3: Simplify the numerator
The numerator simplifies to:
\[ \frac{8}{2 \log_{\sqrt{8}} 8} = \frac{8}{2} \cdot \frac{1}{\log_{\sqrt{8}} 8} = 4 \cdot \frac{1}{\log_{\sqrt{8}} 8} \]

### Step 4: Change of base for \( \log_{\sqrt{8}} 8 \)
We can express \(\log_{\sqrt{8}} 8\) using the change of base formula:
\[ \log_{\sqrt{8}} 8 = \frac{\log_{8} 8}{\log_{8} \sqrt{8}} \]
Since \(\log_{8} 8 = 1\), we need to find \(\log_{8} \sqrt{8}\).

### Step 5: Calculate \( \log_{8} \sqrt{8} \)
We know that \(\sqrt{8} = 8^{1/2}\), so:
\[ \log_{8} \sqrt{8} = \log_{8} (8^{1/2}) = \frac{1}{2} \log_{8} 8 = \frac{1}{2} \cdot 1 = \frac{1}{2} \]

### Step 6: Substitute back into the expression
Now substitute \(\log_{\sqrt{8}} 8\) back into the expression:
\[ \log_{\sqrt{8}} 8 = \frac{1}{\frac{1}{2}} = 2 \]

### Step 7: Final simplification
Now we can substitute this back into our expression:
\[ 4 \cdot \frac{1}{2} = 4 \cdot \frac{1}{2} = 2 \]

### Final Answer
Thus, the value of \(\frac{8 \log_{8} 8}{2 \log_{\sqrt{8}} 8}\) is:
\[ \boxed{2} \]