To expand the given logarithmic expression [tex]\(\log \frac{y^4}{z}\)[/tex] using properties of logarithms, follow these detailed steps:
1. Applying the Quotient Rule:
The quotient rule for logarithms states that:
[tex]\[
\log \left( \frac{a}{b} \right) = \log(a) - \log(b)
\][/tex]
Here, [tex]\(a = y^4\)[/tex] and [tex]\(b = z\)[/tex].
So, we can write:
[tex]\[
\log \frac{y^4}{z} = \log(y^4) - \log(z)
\][/tex]
2. Applying the Power Rule:
The power rule for logarithms states that:
[tex]\[
\log(a^b) = b \cdot \log(a)
\][/tex]
Here, [tex]\(a = y\)[/tex] and [tex]\(b = 4\)[/tex].
So, we can write:
[tex]\[
\log(y^4) = 4 \cdot \log(y)
\][/tex]
3. Substitute Back:
Substitute [tex]\(4 \cdot \log(y)\)[/tex] back into the expression for the logarithm of the quotient:
[tex]\[
\log \frac{y^4}{z} = 4 \cdot \log(y) - \log(z)
\][/tex]
Therefore, the expanded form of [tex]\(\log \frac{y^4}{z}\)[/tex] is:
[tex]\[
4 \log(y) - \log(z)
\][/tex]
