`(log_(8)17)/(log_(9)23) - (log_(2sqrt(2))17)/(log_(3)23)` is equal to

Options

A. 0
B. 1
C. `(17)/(8)`
D. `(23)/(17)`

Answer

To solve the expression \(\frac{\log_{8} 17}{\log_{9} 23} - \frac{\log_{2\sqrt{2}} 17}{\log_{3} 23}\), we will use properties of logarithms to simplify it step by step.

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

We know that:
- \(\log_{8} 17 = \frac{\log_{2} 17}{\log_{2} 8} = \frac{\log_{2} 17}{3}\) (since \(8 = 2^3\))
- \(\log_{9} 23 = \frac{\log_{3} 23}{\log_{3} 9} = \frac{\log_{3} 23}{2}\) (since \(9 = 3^2\))
- \(\log_{2\sqrt{2}} 17 = \frac{\log_{2} 17}{\log_{2} (2\sqrt{2})} = \frac{\log_{2} 17}{\frac{3}{2}} = \frac{2 \log_{2} 17}{3}\) (since \(2\sqrt{2} = 2^{3/2}\))

Now substituting these into the expression gives us:

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

### Step 2: Simplify the fractions.

This can be simplified to:

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

Combining these fractions gives:

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

### Conclusion

Thus, the expression simplifies to:

\[ 0 \]

Therefore, the answer is \(0\).

---