If `log_(10)2=0*30103`, then `log_(10)50` =

Options

A. `2.30103`
B. `2.69897`
C. `1.69897`
D. `0.69897`

Answer

To solve the problem, we need to find \( \log_{10} 50 \) using the given information that \( \log_{10} 2 = 0.30103 \).

### Step-by-Step Solution:

1. **Express 50 in terms of its prime factors**:
\[ 50 = 5 \times 10 = 5 \times (2 \times 5) = 5^2 \times 2 \]

2. **Use the logarithmic property**:
We can express \( \log_{10} 50 \) using the properties of logarithms:
\[ \log_{10} 50 = \log_{10} (5^2 \times 2) \]

3. **Apply the logarithmic properties**:
Using the property \( \log (a \times b) = \log a + \log b \) and \( \log (a^b) = b \log a \):
\[ \log_{10} 50 = \log_{10} (5^2) + \log_{10} 2 = 2 \log_{10} 5 + \log_{10} 2 \]

4. **Find \( \log_{10} 5 \)**:
We know that:
\[ \log_{10} 10 = 1 \]
and since \( 10 = 2 \times 5 \), we can write:
\[ \log_{10} 10 = \log_{10} 2 + \log_{10} 5 \]
Therefore:
\[ 1 = \log_{10} 2 + \log_{10} 5 \]
Rearranging gives:
\[ \log_{10} 5 = 1 - \log_{10} 2 \]

5. **Substituting the value of \( \log_{10} 2 \)**:
Substitute \( \log_{10} 2 = 0.30103 \):
\[ \log_{10} 5 = 1 - 0.30103 = 0.69897 \]

6. **Substituting back into the equation for \( \log_{10} 50 \)**:
Now substitute \( \log_{10} 5 \) back into the equation:
\[ \log_{10} 50 = 2 \log_{10} 5 + \log_{10} 2 \]
\[ \log_{10} 50 = 2(0.69897) + 0.30103 \]

7. **Calculate the final value**:
\[ \log_{10} 50 = 1.39794 + 0.30103 = 1.69897 \]

### Final Answer:
\[ \log_{10} 50 = 1.69897 \]