If `log_(a)m= x`, then `"log"_(1//a) (1)/(m)` is equal to

Options

A. `x`
B. `-x`
C. `(1)/(x)`
D. `(-1)/(x)`

Answer

To solve the problem, we need to find the value of \( \log_{1/a} \left( \frac{1}{m} \right) \) given that \( \log_a m = x \).

### Step-by-step Solution:

1. **Start with the given equation**:
\[ \log_a m = x \]

2. **Use the change of base formula**:
We know that:
\[ \log_a m = \frac{\log m}{\log a} \]
Therefore, we can rewrite the equation as:
\[ \frac{\log m}{\log a} = x \]

3. **Express \( \log m \) in terms of \( \log a \)**:
Rearranging the above equation gives:
\[ \log m = x \cdot \log a \]

4. **Now, we need to find \( \log_{1/a} \left( \frac{1}{m} \right) \)**:
Using the change of base formula again:
\[ \log_{1/a} \left( \frac{1}{m} \right) = \frac{\log \left( \frac{1}{m} \right)}{\log \left( \frac{1}{a} \right)} \]

5. **Calculate \( \log \left( \frac{1}{m} \right) \)**:
We can express this as:
\[ \log \left( \frac{1}{m} \right) = \log 1 - \log m = 0 - \log m = -\log m \]

6. **Calculate \( \log \left( \frac{1}{a} \right) \)**:
Similarly, we have:
\[ \log \left( \frac{1}{a} \right) = \log 1 - \log a = 0 - \log a = -\log a \]

7. **Substituting back into the equation**:
Now substituting these values into our expression:
\[ \log_{1/a} \left( \frac{1}{m} \right) = \frac{-\log m}{-\log a} = \frac{\log m}{\log a} \]

8. **Using the earlier result**:
From step 3, we know that \( \frac{\log m}{\log a} = x \). Therefore:
\[ \log_{1/a} \left( \frac{1}{m} \right) = x \]

### Final Answer:
\[ \log_{1/a} \left( \frac{1}{m} \right) = x \]