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 \( \log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15} \), we can use properties of logarithms. Here’s a step-by-step solution:

### Step 1: Combine the logarithms using the property of addition
Using the property \( \log a + \log b = \log (a \cdot b) \), we can combine the first two logarithms:

\[ \log \frac{14}{3} + \log \frac{11}{5} = \log \left( \frac{14}{3} \cdot \frac{11}{5} \right) \]

### Step 2: Simplify the expression inside the logarithm
Now, we can multiply the fractions:

\[ \frac{14}{3} \cdot \frac{11}{5} = \frac{14 \cdot 11}{3 \cdot 5} = \frac{154}{15} \]

So, we have:

\[ \log \left( \frac{154}{15} \right) - \log \frac{22}{15} \]

### Step 3: Use the property of subtraction
Now we can apply the property \( \log a - \log b = \log \left( \frac{a}{b} \right) \):

\[ \log \left( \frac{154}{15} \right) - \log \left( \frac{22}{15} \right) = \log \left( \frac{154/15}{22/15} \right) \]

### Step 4: Simplify the fraction
The \( 15 \) in the numerator and denominator cancels out:

\[ \frac{154/15}{22/15} = \frac{154}{22} \]

### Step 5: Further simplify \( \frac{154}{22} \)
Now, we can simplify \( \frac{154}{22} \):

\[ \frac{154}{22} = \frac{77}{11} = 7 \]

### Step 6: Write the final logarithmic expression
Thus, we have:

\[ \log 7 \]

### Final Answer
The value of \( \log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15} \) is \( \log 7 \).

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