if `log_(10)x = 1.9675`, then the value of `log_(10)(100x)` is:

Options

A. 196.75
B. 3.9675
C. 2.9675
D. none of these

Answer

To solve the problem, we need to find the value of \( \log_{10}(100x) \) given that \( \log_{10} x = 1.9675 \).

### Step-by-Step Solution:

1. **Use the property of logarithms**:
We can break down \( \log_{10}(100x) \) using the property of logarithms that states \( \log_a(mn) = \log_a(m) + \log_a(n) \).
\[ \log_{10}(100x) = \log_{10}(100) + \log_{10}(x) \]

2. **Calculate \( \log_{10}(100) \)**:
Since \( 100 = 10^2 \), we can use the property \( \log_{10}(10^n) = n \).
\[ \log_{10}(100) = \log_{10}(10^2) = 2 \]

3. **Substitute the known value of \( \log_{10}(x) \)**:
We know from the problem statement that \( \log_{10}(x) = 1.9675 \).
\[ \log_{10}(100x) = 2 + \log_{10}(x) = 2 + 1.9675 \]

4. **Perform the addition**:
Now, we add the two values together.
\[ 2 + 1.9675 = 3.9675 \]

5. **Final answer**:
Therefore, the value of \( \log_{10}(100x) \) is \( 3.9675 \).

### Conclusion:
The answer is \( \log_{10}(100x) = 3.9675 \).