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 start with the given information and apply logarithmic properties step by step.

### Step-by-Step Solution:

1. **Given Information**: We know that \( \log_a m = x \). This means that \( a^x = m \).

2. **Express the Required Logarithm**: We need to find \( \log_{(1/a)} \left( \frac{1}{m} \right) \).

3. **Using the Change of Base Formula**: We can rewrite the logarithm using the change of base formula:
\[ \log_{(1/a)} \left( \frac{1}{m} \right) = \frac{\log \left( \frac{1}{m} \right)}{\log \left( \frac{1}{a} \right)} \]

4. **Simplifying the Logarithms**:
- The logarithm of a reciprocal can be expressed as:
\[ \log \left( \frac{1}{m} \right) = \log 1 - \log m = 0 - \log m = -\log m \]
- Similarly, for \( \log \left( \frac{1}{a} \right) \):
\[ \log \left( \frac{1}{a} \right) = \log 1 - \log a = 0 - \log a = -\log a \]

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

6. **Using the Given Information**: We know from the problem statement that \( \log_a m = x \), which can be expressed as:
\[ \frac{\log m}{\log a} = x \]

7. **Final Result**: Therefore, we conclude that:
\[ \log_{(1/a)} \left( \frac{1}{m} \right) = x \]

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