Solve `log_(6) 9-log_(9) 27 + log_(8)x = log_(64) x - log_(6) 4`..

Options

A. `1//2`
B. `1//4`
C. `1//8`
D. `1//16`

Explanation

To solve the equation \( \log_{6} 9 - \log_{9} 27 + \log_{8} x = \log_{64} x - \log_{6} 4 \), we will follow these steps:

### Step 1: Rewrite the logarithmic expressions
We can use the change of base formula and properties of logarithms to rewrite the terms in the equation.

\[ \log_{9} 27 = \frac{\log 27}{\log 9} = \frac{\log 3^3}{\log 3^2} = \frac{3 \log 3}{2 \log 3} = \frac{3}{2} \]

Thus, we can rewrite the equation as:

\[ \log_{6} 9 - \frac{3}{2} + \log_{8} x = \log_{64} x - \log_{6} 4 \]

### Step 2: Move all logarithmic terms to one side
Rearranging the equation gives:

\[ \log_{6} 9 + \log_{6} 4 - \frac{3}{2} = \log_{64} x - \log_{8} x \]

### Step 3: Use properties of logarithms
We can combine the logarithms on the left side:

\[ \log_{6} (9 \cdot 4) - \frac{3}{2} = \log_{64} x - \log_{8} x \]

Now, calculate \( 9 \cdot 4 = 36 \):

\[ \log_{6} 36 - \frac{3}{2} = \log_{64} x - \log_{8} x \]

### Step 4: Rewrite the right side
Using the change of base formula again, we can express \( \log_{64} x - \log_{8} x \):

\[ \log_{64} x = \frac{\log x}{\log 64} \quad \text{and} \quad \log_{8} x = \frac{\log x}{\log 8} \]

Thus, we have:

\[ \log_{64} x - \log_{8} x = \frac{\log x}{\log 64} - \frac{\log x}{\log 8} \]

### Step 5: Combine the right side
Factoring out \( \log x \):

\[ \log x \left( \frac{1}{\log 64} - \frac{1}{\log 8} \right) \]

### Step 6: Solve for \( x \)
Now we equate both sides:

\[ \log_{6} 36 - \frac{3}{2} = \log x \left( \frac{1}{\log 64} - \frac{1}{\log 8} \right) \]

Calculating \( \log_{6} 36 \):

\[ \log_{6} 36 = \log_{6} (6^2) = 2 \]

So we have:

\[ 2 - \frac{3}{2} = \log x \left( \frac{1}{\log 64} - \frac{1}{\log 8} \right) \]

This simplifies to:

\[ \frac{1}{2} = \log x \left( \frac{1}{\log 64} - \frac{1}{\log 8} \right) \]

### Step 7: Isolate \( \log x \)
Now we can isolate \( \log x \):

\[ \log x = \frac{\frac{1}{2}}{\frac{1}{\log 64} - \frac{1}{\log 8}} \]

### Step 8: Solve for \( x \)
Finally, exponentiate both sides to solve for \( x \):

\[ x = 10^{\log x} \]

### Final Answer
After performing the calculations, we find:

\[ x = \frac{1}{8} \]