Use the properties of logarithms to completely expand [tex]\ln \frac{11 x^8}{w}[/tex]. Do not include any parentheses in your answer.

Note: When entering natural log in your answer, enter lowercase LN as "In". There is no "natural log" button on the Alta keyboard.



Answer :

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]