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 \( y = 2^{\frac{1}{\log_{x} 8}} \), we will follow a series of steps using properties of logarithms.

### Step-by-Step Solution:

1. **Rewrite the logarithm using the change of base formula**:
\[ \log_{x} 8 = \frac{1}{\log_{8} x} \]
Therefore, we can rewrite \( y \) as:
\[ y = 2^{\log_{8} x} \]

**Hint**: Remember that the change of base formula allows us to switch the base of a logarithm.

2. **Express 8 as a power of 2**:
\[ 8 = 2^3 \]
Thus, we can rewrite \( \log_{8} x \) as:
\[ \log_{8} x = \log_{2^3} x \]

**Hint**: Knowing that \( 8 = 2^3 \) helps in simplifying the logarithm.

3. **Apply the logarithmic property**:
Using the property \( \log_{a^b} c = \frac{1}{b} \log_{a} c \), we have:
\[ \log_{2^3} x = \frac{1}{3} \log_{2} x \]
Therefore, substituting this back into our equation for \( y \):
\[ y = 2^{\frac{1}{3} \log_{2} x} \]

**Hint**: This property allows us to simplify logarithms with powers.

4. **Simplify using another logarithmic property**:
We can use the property \( a^{\log_{b} c} = c^{\log_{b} a} \):
\[ y = x^{\frac{1}{3}} \]

**Hint**: This property is useful for switching the base and the exponent in logarithmic expressions.

5. **Solve for \( x \)**:
To isolate \( x \), we can cube both sides:
\[ x = y^3 \]

**Hint**: Cubing both sides helps to eliminate the fractional exponent.

### Final Answer:
Thus, the value of \( x \) is:
\[ x = y^3 \]