Certainly! Let's expand the given logarithmic expression [tex]\(\log \left( \frac{x^7}{y} \right)\)[/tex] using the properties of logarithms:
1. Quotient Rule of Logarithms: The logarithm of a quotient is the difference of the logarithms. This rule states that:
[tex]\[
\log \left( \frac{A}{B} \right) = \log(A) - \log(B)
\][/tex]
Applying this rule, we can separate the logarithms of the numerator and the denominator:
[tex]\[
\log \left( \frac{x^7}{y} \right) = \log(x^7) - \log(y)
\][/tex]
2. Power Rule of Logarithms: The logarithm of an exponentiated value is the exponent times the logarithm of the base. This rule states that:
[tex]\[
\log(A^B) = B \cdot \log(A)
\][/tex]
Applying this rule to the term [tex]\(\log(x^7)\)[/tex], we get:
[tex]\[
\log(x^7) = 7 \cdot \log(x)
\][/tex]
3. Combining the Results: Substitute the expanded form of [tex]\(\log(x^7)\)[/tex] into the expression from step 1:
[tex]\[
\log \left( \frac{x^7}{y} \right) = 7 \cdot \log(x) - \log(y)
\][/tex]
Therefore, the expanded form of [tex]\(\log \left( \frac{x^7}{y} \right)\)[/tex] is:
[tex]\[
7 \cdot \log(x) - \log(y)
\][/tex]
This is the required expansion where each logarithm involves only one variable and does not include any exponents or fractions.
