To condense the logarithmic expression [tex]\(\ln 6 + 6 \ln z - \ln y\)[/tex], we will utilize the properties of logarithms. Specifically, we will use the product rule, the power rule, and the quotient rule for logarithms.
### Steps
1. Start with the given expression:
[tex]\[
\ln 6 + 6 \ln z - \ln y
\][/tex]
2. Apply the power rule to [tex]\(6 \ln z\)[/tex]:
The power rule states that [tex]\(k \ln a = \ln(a^k)\)[/tex]. Applying this rule to the term [tex]\(6 \ln z\)[/tex] gives:
[tex]\[
6 \ln z = \ln(z^6)
\][/tex]
So the expression now becomes:
[tex]\[
\ln 6 + \ln(z^6) - \ln y
\][/tex]
3. Apply the product rule:
The product rule states that [tex]\(\ln a + \ln b = \ln(ab)\)[/tex]. Applying this rule to [tex]\(\ln 6 + \ln(z^6)\)[/tex] gives:
[tex]\[
\ln(6) + \ln(z^6) = \ln(6z^6)
\][/tex]
So now the expression is:
[tex]\[
\ln(6z^6) - \ln y
\][/tex]
4. Apply the quotient rule:
The quotient rule states that [tex]\(\ln a - \ln b = \ln(\frac{a}{b})\)[/tex]. Applying this rule to [tex]\(\ln(6z^6) - \ln y\)[/tex] gives:
[tex]\[
\ln\left(6z^6\right) - \ln(y) = \ln\left(\frac{6z^6}{y}\right)
\][/tex]
Therefore, the condensed logarithmic expression is:
[tex]\[
\boxed{\ln \left(\frac{6 \cdot z^6}{y} \right)}
\][/tex]
