The value of `log (14/3) + log (11/5) - log (22/15)` is:

Options

A. log 77
B. log 11
C. log 7
D. none of these

Answer

To solve the expression \( \log \left( \frac{14}{3} \right) + \log \left( \frac{11}{5} \right) - \log \left( \frac{22}{15} \right) \), we can use the properties of logarithms.

### Step-by-Step Solution:

1. **Apply the logarithm property for addition**:
\[ \log a + \log b = \log (a \cdot b) \]
Here, we can combine the first two logarithmic terms:
\[ \log \left( \frac{14}{3} \right) + \log \left( \frac{11}{5} \right) = \log \left( \frac{14}{3} \cdot \frac{11}{5} \right) \]

2. **Multiply the fractions**:
\[ \frac{14}{3} \cdot \frac{11}{5} = \frac{14 \cdot 11}{3 \cdot 5} = \frac{154}{15} \]
So now we have:
\[ \log \left( \frac{154}{15} \right) \]

3. **Apply the logarithm property for subtraction**:
\[ \log a - \log b = \log \left( \frac{a}{b} \right) \]
We can now subtract the third logarithmic term:
\[ \log \left( \frac{154}{15} \right) - \log \left( \frac{22}{15} \right) = \log \left( \frac{\frac{154}{15}}{\frac{22}{15}} \right) \]

4. **Simplify the fraction**:
\[ \frac{\frac{154}{15}}{\frac{22}{15}} = \frac{154}{15} \cdot \frac{15}{22} = \frac{154}{22} \]

5. **Simplify \( \frac{154}{22} \)**:
\[ \frac{154 \div 22}{22 \div 22} = \frac{7}{1} = 7 \]

6. **Final logarithm**:
\[ \log(7) \]

Thus, the final answer is:
\[ \log(7) \]