`log_yx=?`

Options

A. `xlog_ey`
B. `x/(log_ey)`
C. `(log_ex)/(log_ey)`
D. `(log_ey)/(log_ex)`

Answer

To solve the problem \( \log_y x \), we can use the change of base formula for logarithms. The change of base formula states that:

\[ \log_b a = \frac{\log_k a}{\log_k b} \]

where \( k \) can be any positive number (commonly chosen as 10 or \( e \)). In this case, we will apply the formula to find \( \log_y x \).

### Step-by-step Solution:

1. **Identify the logarithm to be converted:**
We need to convert \( \log_y x \) using the change of base formula.

2. **Apply the change of base formula:**
Using the formula, we can express \( \log_y x \) as:
\[ \log_y x = \frac{\log_k x}{\log_k y} \]
Here, \( k \) can be any base. For simplicity, we can choose \( k = e \) (natural logarithm) or \( k = 10 \) (common logarithm).

3. **Final expression:**
Thus, we can write:
\[ \log_y x = \frac{\log_e x}{\log_e y} \quad \text{or} \quad \log_y x = \frac{\log_{10} x}{\log_{10} y} \]
Both expressions are valid and represent the same quantity.

### Final Answer:
\[ \log_y x = \frac{\log_e x}{\log_e y} \quad \text{or} \quad \log_y x = \frac{\log_{10} x}{\log_{10} y} \]