What is the value of `2log_(8)2-(1)/(3)log_(3)9`?

Options

A. 0
B. 1
C. 2
D. `1//3`

Explanation

To solve the expression \( 2\log_{8}2 - \frac{1}{3}\log_{3}9 \), we will follow these steps:

### Step 1: Convert the logarithm base
We can use the change of base formula for logarithms. The change of base formula states that:
\[ \log_{a}b = \frac{\log_{c}b}{\log_{c}a} \]
We will convert \( \log_{8}2 \) and \( \log_{3}9 \) to base 2.

### Step 2: Calculate \( \log_{8}2 \)
Using the change of base formula:
\[ \log_{8}2 = \frac{\log_{2}2}{\log_{2}8} \]
Since \( \log_{2}2 = 1 \) and \( \log_{2}8 = 3 \) (because \( 8 = 2^3 \)), we have:
\[ \log_{8}2 = \frac{1}{3} \]

### Step 3: Calculate \( \log_{3}9 \)
Similarly, we can calculate \( \log_{3}9 \):
\[ \log_{3}9 = \frac{\log_{2}9}{\log_{2}3} \]
Since \( 9 = 3^2 \), we can also write:
\[ \log_{3}9 = 2 \]

### Step 4: Substitute back into the expression
Now we can substitute back into the original expression:
\[ 2\log_{8}2 - \frac{1}{3}\log_{3}9 = 2 \cdot \frac{1}{3} - \frac{1}{3} \cdot 2 \]
This simplifies to:
\[ \frac{2}{3} - \frac{2}{3} \]

### Step 5: Final calculation
Now we can perform the final calculation:
\[ \frac{2}{3} - \frac{2}{3} = 0 \]

### Conclusion
Thus, the value of \( 2\log_{8}2 - \frac{1}{3}\log_{3}9 \) is:
\[ \boxed{0} \]