The value of `log_(-1//5) 625` is:

Options

A. `1/4`
B. `-5`
C. `-4`
D. `-3`

Answer

To solve the question `log_(-1/5) 625`, we will follow the properties of logarithms step by step.

### Step-by-Step Solution:

1. **Identify the Base and Argument**:
We have the logarithm with base `-1/5` and the argument `625`.

\[ \text{We need to find } \log_{(-1/5)}(625) \]

2. **Convert the Base**:
We can express the base `-1/5` as `5^{-1}`. This means we can rewrite the logarithm as:

\[ \log_{(-1/5)}(625) = \log_{(5^{-1})}(625) \]

3. **Apply the Change of Base Formula**:
Using the property of logarithms, we know that:

\[ \log_{(a^b)}(c) = \frac{1}{b} \log_a(c) \]

Applying this property here gives us:

\[ \log_{(5^{-1})}(625) = \frac{1}{-1} \log_{5}(625) = -\log_{5}(625) \]

4. **Factor 625**:
Next, we need to express `625` in terms of base `5`. We know that:

\[ 625 = 5^4 \]

5. **Calculate the Logarithm**:
Now we can substitute `625` back into the logarithm:

\[ -\log_{5}(625) = -\log_{5}(5^4) \]

Using the property of logarithms that states `log_a(a^b) = b`, we have:

\[ -\log_{5}(5^4) = -4 \]

6. **Final Result**:
Therefore, the value of `log_{(-1/5)}(625)` is:

\[ \boxed{-4} \]