The value of `log_(8)17/(log_(9)23)-log_(2sqrt2)17/(log_(3)23)` is equal to

Options

A. -1
B. 0
C. `(log_(2)17)/(log_(3)23)`
D. `(4(log_(2)17))/(3(log_(3)23))`

Answer

To solve the expression \( \frac{\log_{8} 17}{\log_{9} 23} - \frac{\log_{2\sqrt{2}} 17}{\log_{3} 23} \), we will use the change of base formula and properties of logarithms.

### Step 1: Rewrite the logarithms using the change of base formula
Using the change of base formula, we can express the logarithms in terms of natural logarithms (or logarithms of any common base):

\[ \log_{a} b = \frac{\log b}{\log a} \]

Thus, we can rewrite the terms:

\[ \log_{8} 17 = \frac{\log 17}{\log 8} = \frac{\log 17}{\log(2^3)} = \frac{\log 17}{3 \log 2} \]

\[ \log_{9} 23 = \frac{\log 23}{\log 9} = \frac{\log 23}{\log(3^2)} = \frac{\log 23}{2 \log 3} \]

\[ \log_{2\sqrt{2}} 17 = \frac{\log 17}{\log(2\sqrt{2})} = \frac{\log 17}{\log(2^{3/2})} = \frac{\log 17}{\frac{3}{2} \log 2} = \frac{2 \log 17}{3 \log 2} \]

\[ \log_{3} 23 = \frac{\log 23}{\log 3} \]

### Step 2: Substitute these into the expression
Now we substitute these values back into the original expression:

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

### Step 3: Simplify the fractions
We simplify each term:

1. The first term becomes:
\[ \frac{\log 17}{3 \log 2} \cdot \frac{2 \log 3}{\log 23} = \frac{2 \log 17 \log 3}{3 \log 2 \log 23} \]

2. The second term becomes:
\[ \frac{2 \log 17}{3 \log 2} \cdot \frac{\log 3}{\log 23} = \frac{2 \log 17 \log 3}{3 \log 2 \log 23} \]

### Step 4: Combine the terms
Now we combine the two terms:
\[ \frac{2 \log 17 \log 3}{3 \log 2 \log 23} - \frac{2 \log 17 \log 3}{3 \log 2 \log 23} = 0 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0} \]