`log_(y)x`=?

Options

A. `x log_(e)y`
B. `x/(log_(e)y)`
C. `(log_(e)x)/(log_(e)y)`
D. `(log_(e)y)/(log_(e)x)`

Answer

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

\[ \log_a b = \frac{\log_c b}{\log_c a} \]

where \( c \) can be any positive number (commonly 10 or \( e \)). In this case, we will use base \( e \) (natural logarithm) for our calculation.

### Step-by-Step Solution:

1. **Apply the Change of Base Formula:**
\[ \log_y x = \frac{\log_e x}{\log_e y} \]

2. **Identify the Components:**
- Here, \( \log_e x \) is the natural logarithm of \( x \).
- \( \log_e y \) is the natural logarithm of \( y \).

3. **Final Expression:**
Thus, we can express \( \log_y x \) as:
\[ \log_y x = \frac{\ln x}{\ln y} \]

### Final Answer:
\[ \log_y x = \frac{\ln x}{\ln y} \]

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