To expand the given logarithmic expression [tex]\(\log \left(\frac{x^7 y^5}{z^9}\right)\)[/tex], we can use the properties of logarithms. Let's go through the steps in detail.
1. Property of the logarithm with a quotient inside: The logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator.
[tex]\[
\log \left(\frac{a}{b}\right) = \log(a) - \log(b)
\][/tex]
Applying this property to the given expression, we get:
[tex]\[
\log \left(\frac{x^7 y^5}{z^9}\right) = \log(x^7 y^5) - \log(z^9)
\][/tex]
2. Property of the logarithm with a product inside: The logarithm of a product is equal to the sum of the logarithms of the factors.
[tex]\[
\log(ab) = \log(a) + \log(b)
\][/tex]
Applying this property to [tex]\(\log(x^7 y^5)\)[/tex], we get:
[tex]\[
\log(x^7 y^5) = \log(x^7) + \log(y^5)
\][/tex]
3. Property of the logarithm with an exponent inside: The logarithm of a power is equal to the exponent times the logarithm of the base.
[tex]\[
\log(a^n) = n \log(a)
\][/tex]
Applying this property to both [tex]\(\log(x^7)\)[/tex] and [tex]\(\log(y^5)\)[/tex], we get:
[tex]\[
\log(x^7) = 7 \log(x)
\][/tex]
[tex]\[
\log(y^5) = 5 \log(y)
\][/tex]
Similarly, applying this property to [tex]\(\log(z^9)\)[/tex], we get:
[tex]\[
\log(z^9) = 9 \log(z)
\][/tex]
4. Combining all parts together: Now we combine all the parts we have found.
[tex]\[
\log(x^7 y^5) - \log(z^9) = \left(7 \log(x) + 5 \log(y)\right) - 9 \log(z)
\][/tex]
So, the expanded form of the given logarithmic expression is:
[tex]\[
7 \log(x) + 5 \log(y) - 9 \log(z)
\][/tex]
