To write the given expression [tex]\( 7 \ln (x) + 3 \ln (y) - 6 \ln (z) \)[/tex] as a single logarithm, we need to use the properties of logarithms, specifically the power rule and the product/quotient rule.
1. Power Rule: This rule states that [tex]\( a \ln(b) = \ln(b^a) \)[/tex]. We can apply this rule to each term in the expression to move the coefficients inside the logarithms as exponents.
[tex]\[
7 \ln(x) = \ln(x^7)
\][/tex]
[tex]\[
3 \ln(y) = \ln(y^3)
\][/tex]
[tex]\[
-6 \ln(z) = \ln(z^{-6})
\][/tex]
2. Product/Quotient Rule: This rule states that [tex]\( \ln(a) + \ln(b) = \ln(ab) \)[/tex] and [tex]\( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \)[/tex]. We can apply these rules to combine the logarithms.
First, combine the logarithms involving addition:
[tex]\[
\ln(x^7) + \ln(y^3) = \ln(x^7 y^3)
\][/tex]
Next, incorporate the subtraction:
[tex]\[
\ln(x^7 y^3) - \ln(z^{-6}) = \ln\left(\frac{x^7 y^3}{z^{-6}}\right)
\][/tex]
Since [tex]\( z^{-6} = \frac{1}{z^6} \)[/tex], we rewrite the quotient as:
[tex]\[
\frac{x^7 y^3}{z^{-6}} = x^7 y^3 \cdot z^6
\][/tex]
3. Combine all terms: Now we can combine the terms into a single logarithm:
[tex]\[
\ln\left(x^7 y^3 z^6\right)
\][/tex]
Thus, the original expression [tex]\( 7 \ln (x) + 3 \ln (y) - 6 \ln (z) \)[/tex] can be written as:
[tex]\[
\boxed{\ln\left(x^7 y^3 z^6\right)}
\][/tex]
