If ` y=2^(1/log_x 8)`, then x is equal to

Options

A. a) y
B. b) `y^(2)`
C. c) `y^(3)`
D. 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
We start with the equation:
\[ y = 2^{\frac{1}{\log_x 8}} \]
Using the change of base formula for logarithms, we can rewrite \( \log_x 8 \) as:
\[ \log_x 8 = \frac{\log_2 8}{\log_2 x} \]
Thus, we can rewrite the equation as:
\[ y = 2^{\frac{\log_2 x}{\log_2 8}} \]

### Step 2: Simplify \( \log_2 8 \)
Next, we know that \( 8 = 2^3 \), so:
\[ \log_2 8 = 3 \]
Substituting this back into our equation gives:
\[ y = 2^{\frac{\log_2 x}{3}} \]

### Step 3: Rewrite the equation
We can express this as:
\[ y = 2^{\log_2 x^{\frac{1}{3}}} \]

### Step 4: Remove the base 2
Since the bases are the same, we can equate the exponents:
\[ y = x^{\frac{1}{3}} \]

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

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{y^3} \]