`(log_(8)17)/(log_(9)23)-(log_(2sqrt(2))17)/(log_(3)23)=`

Options

A. `(17)/(8)`
B. `2/3`
C. `8/9`
D. 0

Answer

To solve the expression

\[ \frac{\log_{8} 17}{\log_{9} 23} - \frac{\log_{2\sqrt{2}} 17}{\log_{3} 23} \]

we will simplify each logarithm using the change of base formula and properties of logarithms.

### Step 1: Rewrite the logarithms in terms of base 2 and base 3

1. **Convert \(\log_{8} 17\)**:
\[ \log_{8} 17 = \log_{2^3} 17 = \frac{1}{3} \log_{2} 17 \]

2. **Convert \(\log_{9} 23\)**:
\[ \log_{9} 23 = \log_{3^2} 23 = \frac{1}{2} \log_{3} 23 \]

3. **Convert \(\log_{2\sqrt{2}} 17\)**:
\[ \log_{2\sqrt{2}} 17 = \log_{2^{3/2}} 17 = \frac{2}{3} \log_{2} 17 \]

4. **Convert \(\log_{3} 23\)** (remains the same):
\[ \log_{3} 23 \]

### Step 2: Substitute these values back into the expression

Now substituting these into the original expression:

\[ \frac{\frac{1}{3} \log_{2} 17}{\frac{1}{2} \log_{3} 23} - \frac{\frac{2}{3} \log_{2} 17}{\log_{3} 23} \]

### Step 3: Simplify each term

1. **First term**:
\[ \frac{\frac{1}{3} \log_{2} 17}{\frac{1}{2} \log_{3} 23} = \frac{1}{3} \cdot \frac{2}{1} \cdot \frac{\log_{2} 17}{\log_{3} 23} = \frac{2}{3} \cdot \frac{\log_{2} 17}{\log_{3} 23} \]

2. **Second term**:
\[ \frac{\frac{2}{3} \log_{2} 17}{\log_{3} 23} \]

### Step 4: Combine the terms

Now we can combine the two terms:

\[ \frac{2}{3} \cdot \frac{\log_{2} 17}{\log_{3} 23} - \frac{2}{3} \cdot \frac{\log_{2} 17}{\log_{3} 23} = 0 \]

### Final Answer

Thus, the value of the expression is

\[ \boxed{0} \]