`log_(0.01) 1000 +log_(0.1)0.0001` is equal to :

Options

A. -2
B. 3
C. `-5//2`
D. `5//2`

Answer

To solve the expression \( \log_{0.01} 1000 + \log_{0.1} 0.0001 \), we will break it down step by step.

### Step 1: Rewrite the logarithms using the change of base formula
We can use the change of base formula for logarithms, which states that \( \log_a b = \frac{\log_{10} b}{\log_{10} a} \).

So, we can rewrite:
\[ \log_{0.01} 1000 = \frac{\log_{10} 1000}{\log_{10} 0.01} \]
and
\[ \log_{0.1} 0.0001 = \frac{\log_{10} 0.0001}{\log_{10} 0.1} \]

### Step 2: Calculate the logarithms
Now we will calculate the logarithms:
- \( \log_{10} 1000 = \log_{10} (10^3) = 3 \)
- \( \log_{10} 0.01 = \log_{10} (10^{-2}) = -2 \)
- \( \log_{10} 0.0001 = \log_{10} (10^{-4}) = -4 \)
- \( \log_{10} 0.1 = \log_{10} (10^{-1}) = -1 \)

### Step 3: Substitute the values back into the expression
Now substituting these values into our expressions:
\[ \log_{0.01} 1000 = \frac{3}{-2} = -\frac{3}{2} \]
and
\[ \log_{0.1} 0.0001 = \frac{-4}{-1} = 4 \]

### Step 4: Combine the results
Now we can combine the results:
\[ \log_{0.01} 1000 + \log_{0.1} 0.0001 = -\frac{3}{2} + 4 \]

### Step 5: Simplify the expression
To simplify \( -\frac{3}{2} + 4 \):
- Convert 4 to a fraction: \( 4 = \frac{8}{2} \)
- Now combine:
\[ -\frac{3}{2} + \frac{8}{2} = \frac{8 - 3}{2} = \frac{5}{2} \]

### Final Answer
Thus, the final answer is:
\[ \frac{5}{2} \]