If `log_(10) 2 = 0.030103, log_(10) 50 = `

Options

A. 2.30103
B. 2.698987
C. 1.99897
D. 0.69897

Answer

To find \( \log_{10} 50 \) given that \( \log_{10} 2 = 0.030103 \), we can use the properties of logarithms.

### Step-by-Step Solution:

1. **Express 50 in terms of its factors**:
\[ 50 = 5 \times 10 \]

2. **Use the logarithm property**:
\[ \log_{10} 50 = \log_{10} (5 \times 10) = \log_{10} 5 + \log_{10} 10 \]

3. **Substitute the value of \( \log_{10} 10 \)**:
\[ \log_{10} 10 = 1 \]
So, we have:
\[ \log_{10} 50 = \log_{10} 5 + 1 \]

4. **Express 5 in terms of 2**:
\[ 5 = \frac{10}{2} \]
Therefore:
\[ \log_{10} 5 = \log_{10} \left(\frac{10}{2}\right) = \log_{10} 10 - \log_{10} 2 \]

5. **Substitute \( \log_{10} 10 \) and \( \log_{10} 2 \)**:
\[ \log_{10} 5 = 1 - \log_{10} 2 = 1 - 0.030103 \]

6. **Calculate \( \log_{10} 5 \)**:
\[ \log_{10} 5 = 1 - 0.030103 = 0.969897 \]

7. **Substitute back to find \( \log_{10} 50 \)**:
\[ \log_{10} 50 = 0.969897 + 1 = 1.969897 \]

### Final Answer:
\[ \log_{10} 50 \approx 1.969897 \]