`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 can use the properties of logarithms.

### Step 1: Rewrite the logarithms in terms of base 10
We know that:
- \( 0.01 = 10^{-2} \)
- \( 1000 = 10^3 \)
- \( 0.1 = 10^{-1} \)
- \( 0.0001 = 10^{-4} \)

Thus, we can rewrite the logarithms:
\[ \log_{0.01} 1000 = \log_{10^{-2}} 10^3 \]
\[ \log_{0.1} 0.0001 = \log_{10^{-1}} 10^{-4} \]

### Step 2: Apply the change of base formula
Using the change of base formula \( \log_{a^m} b^n = \frac{n}{m} \log_{a} b \), we can simplify:
\[ \log_{10^{-2}} 10^3 = \frac{3}{-2} \log_{10} 10 = \frac{3}{-2} \cdot 1 = -\frac{3}{2} \]
\[ \log_{10^{-1}} 10^{-4} = \frac{-4}{-1} \log_{10} 10 = 4 \cdot 1 = 4 \]

### Step 3: Combine the results
Now we can add the two results together:
\[ -\frac{3}{2} + 4 \]

### Step 4: Convert 4 to a fraction
To add these, we convert 4 to a fraction with a denominator of 2:
\[ 4 = \frac{8}{2} \]

### Step 5: Perform the addition
Now we can add:
\[ -\frac{3}{2} + \frac{8}{2} = \frac{8 - 3}{2} = \frac{5}{2} \]

### Final Result
Thus, the value of \( \log_{0.01} 1000 + \log_{0.1} 0.0001 \) is:
\[ \frac{5}{2} \]

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