To expand the expression [tex]\(\log \frac{y^4}{z}\)[/tex] using the properties of logarithms, follow these steps:
### Step 1: Use the Division Rule of Logarithms
The division rule of logarithms states:
[tex]\[
\log \left( \frac{a}{b} \right) = \log(a) - \log(b)
\][/tex]
Applying this rule to [tex]\(\log \frac{y^4}{z}\)[/tex], we get:
[tex]\[
\log \frac{y^4}{z} = \log(y^4) - \log(z)
\][/tex]
### Step 2: Use the Power Rule of Logarithms
The power rule of logarithms states:
[tex]\[
\log (a^b) = b \cdot \log(a)
\][/tex]
Applying this rule to [tex]\(\log(y^4)\)[/tex], we get:
[tex]\[
\log(y^4) = 4 \cdot \log(y)
\][/tex]
### Step 3: Combine the Terms
Now, substitute [tex]\(\log(y^4)\)[/tex] back into the expression we obtained from the division rule:
[tex]\[
\log \frac{y^4}{z} = 4 \cdot \log(y) - \log(z)
\][/tex]
### Result
The expanded form using properties of logarithms is:
[tex]\[
\log \frac{y^4}{z} = 4 \log(y) - \log(z)
\][/tex]
Each term now involves only one variable and does not have any exponents or fractions.
Hence, the final expanded form is:
[tex]\[
\log \frac{y^4}{z} = 4 \log(y) - \log(z)
\][/tex]
