Sure! Let's use the properties of logarithms to expand [tex]\(\log \frac{y}{z^6}\)[/tex] step by step.
### Step 1: Applying the Quotient Rule
The quotient rule for logarithms states that [tex]\(\log \left(\frac{a}{b}\right) = \log(a) - \log(b)\)[/tex]. Applying this rule to [tex]\(\log \frac{y}{z^6}\)[/tex], we get:
[tex]\[
\log \left(\frac{y}{z^6}\right) = \log(y) - \log(z^6)
\][/tex]
### Step 2: Applying the Power Rule
Next, we apply the power rule for logarithms, which states that [tex]\(\log(a^b) = b \log(a)\)[/tex]. Here, [tex]\(b = 6\)[/tex] and [tex]\(a = z\)[/tex], so we apply the power rule to [tex]\(\log(z^6)\)[/tex]:
[tex]\[
\log(z^6) = 6 \log(z)
\][/tex]
### Step 3: Substitution
Now, substitute [tex]\(6 \log(z)\)[/tex] back into the previous expression:
[tex]\[
\log(y) - \log(z^6) = \log(y) - 6 \log(z)
\][/tex]
### Conclusion
The expanded form of [tex]\(\log \frac{y}{z^6}\)[/tex] is:
[tex]\[
\log(y) - 6 \log(z)
\][/tex]
Therefore, [tex]\(\log \frac{y}{z^6} = \log(y) - 6 \log(z)\)[/tex].
