To solve the expression [tex]\(\log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15}\)[/tex], we can use properties of logarithms to combine these logarithmic terms into a single logarithmic expression.
First, recall the logarithm properties:
1. [tex]\(\log a + \log b = \log (a \times b)\)[/tex]
2. [tex]\(\log a - \log b = \log \left( \frac{a}{b} \right)\)[/tex]
Using these properties, we can simplify the expression step by step.
Starting with:
[tex]\[
\log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15}
\][/tex]
Step 1: Combine the sums using the property [tex]\(\log a + \log b = \log (a \times b)\)[/tex]:
[tex]\[
\log \frac{14}{3} + \log \frac{11}{5} = \log \left( \frac{14}{3} \times \frac{11}{5} \right)
\][/tex]
Step 2: Simplify the product inside the logarithm:
[tex]\[
\frac{14}{3} \times \frac{11}{5} = \frac{14 \times 11}{3 \times 5} = \frac{154}{15}
\][/tex]
Now we have:
[tex]\[
\log \left( \frac{154}{15} \right) - \log \frac{22}{15}
\][/tex]
Step 3: Combine the difference using the property [tex]\(\log a - \log b = \log \left( \frac{a}{b} \right)\)[/tex]:
[tex]\[
\log \left( \frac{154}{15} \right) - \log \frac{22}{15} = \log \left( \frac{\frac{154}{15}}{\frac{22}{15}} \right)
\][/tex]
Step 4: Simplify the fraction inside the logarithm:
[tex]\[
\frac{\frac{154}{15}}{\frac{22}{15}} = \frac{154}{15} \times \frac{15}{22} = \frac{154 \times 15}{15 \times 22} = \frac{154}{22} = 7
\][/tex]
So, the whole original expression simplifies to:
[tex]\[
\log 7
\][/tex]
Therefore, the correct answer is:
[tex]\[
\log 7
\][/tex]
