If `x^(y) =k`, then

Options

A. `log_(y)k =x`
B. `log_(x), y=k`
C. `log_(k)x =y`
D. `log_(x)k=y`

Answer

To solve the equation \( x^y = k \), we can use logarithmic properties to express \( y \) in terms of \( k \) and \( x \). Let's break down the steps:

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x^y = k \]

2. **Take the logarithm of both sides**:
We can take the logarithm of both sides of the equation. For simplicity, we will use the natural logarithm (though any logarithm base will work):
\[ \log(x^y) = \log(k) \]

3. **Apply the power rule of logarithms**:
According to the power rule of logarithms, \( \log(a^b) = b \cdot \log(a) \). Thus, we can rewrite the left side:
\[ y \cdot \log(x) = \log(k) \]

4. **Solve for \( y \)**:
To isolate \( y \), divide both sides by \( \log(x) \):
\[ y = \frac{\log(k)}{\log(x)} \]

5. **Use the change of base formula**:
The change of base formula states that \( \log_a(b) = \frac{\log_c(b)}{\log_c(a)} \). Therefore, we can express \( y \) as:
\[ y = \log_x(k) \]

### Conclusion:
Thus, we have derived that:
\[ y = \log_x(k) \]