The value of `log_(0.001) (10000)` is:

Options

A. `1//2`
B. `-2`
C. `1//4`
D. `-4`

Answer

To solve the question `log_(0.001) (10000)`, we can follow these steps:

### Step 1: Rewrite the logarithm in terms of base 10
We can use the change of base formula for logarithms, which states:
\[ \log_a(b) = \frac{\log_c(b)}{\log_c(a)} \]
In this case, we can choose base 10:
\[ \log_{0.001}(10000) = \frac{\log_{10}(10000)}{\log_{10}(0.001)} \]

### Step 2: Calculate \(\log_{10}(10000)\)
We know that:
\[ 10000 = 10^4 \]
Thus:
\[ \log_{10}(10000) = 4 \]

### Step 3: Calculate \(\log_{10}(0.001)\)
We can express \(0.001\) as:
\[ 0.001 = 10^{-3} \]
Thus:
\[ \log_{10}(0.001) = -3 \]

### Step 4: Substitute the values into the change of base formula
Now we substitute the values we found into the formula:
\[ \log_{0.001}(10000) = \frac{4}{-3} \]

### Step 5: Simplify the expression
This simplifies to:
\[ \log_{0.001}(10000) = -\frac{4}{3} \]

### Final Answer
Thus, the value of \( \log_{0.001}(10000) \) is:
\[ -\frac{4}{3} \]

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