Sure, let's solve the given logarithmic expression step by step.
The expression to solve is:
[tex]\[
\log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15}
\][/tex]
1. Use the property of logarithms:
The first property we can employ is that the sum of logarithms is equal to the logarithm of the product:
[tex]\[
\log a + \log b = \log (a \cdot b)
\][/tex]
Applying this property to the first two terms:
[tex]\[
\log \frac{14}{3} + \log \frac{11}{5} = \log \left(\frac{14}{3} \cdot \frac{11}{5}\right)
\][/tex]
2. Simplify the multiplication inside the logarithm:
Perform the multiplication inside the logarithm:
[tex]\[
\frac{14}{3} \cdot \frac{11}{5} = \frac{14 \cdot 11}{3 \cdot 5} = \frac{154}{15}
\][/tex]
So the expression now is:
[tex]\[
\log \left(\frac{154}{15}\right)
\][/tex]
3. Use another logarithmic property:
The property for subtraction of logarithms is:
[tex]\[
\log a - \log b = \log \left(\frac{a}{b}\right)
\][/tex]
Apply this property to the expression:
[tex]\[
\log \left(\frac{154}{15}\right) - \log \left(\frac{22}{15}\right) = \log \left(\frac{\frac{154}{15}}{\frac{22}{15}}\right)
\][/tex]
4. Simplify the fraction inside the logarithm:
Simplify the division of fractions:
[tex]\[
\frac{\frac{154}{15}}{\frac{22}{15}} = \frac{154}{15} \cdot \frac{15}{22} = \frac{154 \cdot 15}{15 \cdot 22} = \frac{154}{22}
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
\log \left(\frac{154}{22}\right)
\][/tex]
5. Further simplify the fraction:
Simplify the fraction inside the logarithm:
[tex]\[
\frac{154}{22} = \frac{154 \div 22}{22 \div 22} = \frac{7}{1} = 7
\][/tex]
So the final result is:
[tex]\[
\log 7
\][/tex]
Therefore:
[tex]\[
\log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15} = \log 7
\][/tex]
The final simplified answer is:
[tex]\[
\log 7
\][/tex]
