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Simplify the expression:

[tex]\[
\log \frac{14}{3}+\log \frac{11}{5}-\log \frac{22}{15}=\log
\][/tex]


Sagot :

To solve the given expression involving logarithms, we will use the properties of logarithms to simplify it:

1. The initial expression is:
[tex]\[ \log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15} \][/tex]

2. Use the property of logarithms that states [tex]\(\log a + \log b = \log (a \cdot b)\)[/tex]:
[tex]\[ \log \left( \frac{14}{3} \cdot \frac{11}{5} \right) - \log \frac{22}{15} \][/tex]

3. Multiply the fractions inside the logarithm:
[tex]\[ \frac{14}{3} \cdot \frac{11}{5} = \frac{14 \cdot 11}{3 \cdot 5} = \frac{154}{15} \][/tex]

4. Substitute back into the expression:
[tex]\[ \log \frac{154}{15} - \log \frac{22}{15} \][/tex]

5. Use the property [tex]\(\log a - \log b = \log \left( \frac{a}{b} \right)\)[/tex]:
[tex]\[ \log \left( \frac{\frac{154}{15}}{\frac{22}{15}} \right) \][/tex]

6. Simplify the fraction inside the logarithm:
[tex]\[ \frac{\frac{154}{15}}{\frac{22}{15}} = \frac{154}{15} \cdot \frac{15}{22} = \frac{154}{22} = 7 \][/tex]

7. Substitute back into the expression:
[tex]\[ \log 7 \][/tex]

Thus, the correct answer is:
[tex]\[ \log 7 \][/tex]
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