Use the properties of logarithms to expand [tex]\log \frac{y^4}{z}[/tex].

Each logarithm should involve only one variable and should not have any exponents or fractions. Assume that all variables are positive.

[tex]\log \frac{y^4}{z} = 4 \log y - \log z[/tex]



Answer :

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]