If `y = 2^(1//"log"_(x)8)`, then x equal to

Options

A. y
B. `y^(2)`
C. `y^(3)`
D. none of these

Answer

To solve the equation \( y = 2^{\frac{1}{\log_x 8}} \) for \( x \), we can follow these steps:

### Step 1: Rewrite the logarithm
Using the change of base formula for logarithms, we can rewrite \( \log_x 8 \) as:
\[ \log_x 8 = \frac{\log 8}{\log x} \]
Thus, we can express \( y \) as:
\[ y = 2^{\frac{1}{\frac{\log 8}{\log x}}} = 2^{\frac{\log x}{\log 8}} \]

### Step 2: Simplify the exponent
Now we can rewrite the expression:
\[ y = 2^{\log_x 8} = (2^{\log 8})^{\frac{1}{\log x}} \]
Using the property \( a^{\log_b c} = c^{\log_b a} \), we can express \( 2^{\log 8} \):
\[ 2^{\log 8} = 8^{\log 2} \]
Thus, we have:
\[ y = 8^{\frac{\log 2}{\log x}} \]

### Step 3: Set up the equation
Now we can equate the two expressions:
\[ y = 8^{\frac{\log 2}{\log x}} \]
Taking the logarithm of both sides gives:
\[ \log y = \frac{\log 2}{\log x} \cdot \log 8 \]

### Step 4: Solve for \( \log x \)
Rearranging the equation to isolate \( \log x \):
\[ \log x = \frac{\log 2 \cdot \log 8}{\log y} \]

### Step 5: Substitute \( \log 8 \)
Since \( \log 8 = 3 \log 2 \), we can substitute:
\[ \log x = \frac{\log 2 \cdot 3 \log 2}{\log y} = \frac{3 (\log 2)^2}{\log y} \]

### Step 6: Solve for \( x \)
Now, exponentiating both sides gives:
\[ x = 10^{\frac{3 (\log 2)^2}{\log y}} \]
This is the value of \( x \) in terms of \( y \).

### Final Answer
The final expression for \( x \) is:
\[ x = 2^{3} = 8 \quad \text{(if } y = 2 \text{)} \]