To expand the logarithm [tex]\(\ln \frac{11 x^8}{w}\)[/tex] using the properties of logarithms, we follow these steps:
1. Quotient Rule of Logarithms: Recall the property that [tex]\(\ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b)\)[/tex].
Applying this property, we write:
[tex]\[
\ln \frac{11 x^8}{w} = \ln (11 x^8) - \ln (w)
\][/tex]
2. Product Rule of Logarithms: Recall the property that [tex]\(\ln(a \cdot b) = \ln(a) + \ln(b)\)[/tex].
Applying this property to [tex]\(\ln (11 x^8)\)[/tex], we write:
[tex]\[
\ln (11 x^8) = \ln (11) + \ln (x^8)
\][/tex]
3. Power Rule of Logarithms: Recall the property that [tex]\(\ln (a^b) = b \cdot \ln (a)\)[/tex].
Applying this property to [tex]\(\ln (x^8)\)[/tex], we write:
[tex]\[
\ln (x^8) = 8 \cdot \ln (x)
\][/tex]
Combining these results, our expression becomes:
[tex]\[
\ln \frac{11 x^8}{w} = \ln (11) + \ln (x^8) - \ln (w)
\][/tex]
[tex]\[
\ln \frac{11 x^8}{w} = \ln (11) + 8 \cdot \ln (x) - \ln (w)
\][/tex]
Expressing natural logarithm as "In", we get:
[tex]\[
\ln \frac{11 x^8}{w} = In (11) + 8 \cdot In (x) - In (w)
\][/tex]
Thus, the completely expanded form of [tex]\(\ln \frac{11 x^8}{w}\)[/tex] is:
[tex]\[
In (11) + 8 \cdot In (x) - In (w)
\][/tex]
