To expand the logarithmic expression [tex]\(\ln\left(\frac{5 x^2 y^4}{w^3}\right)\)[/tex], we can use the properties of logarithms. Specifically, we'll apply the following logarithmic properties:
1. [tex]\(\ln(ab) = \ln(a) + \ln(b)\)[/tex]
2. [tex]\(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)[/tex]
3. [tex]\(\ln(a^n) = n \cdot \ln(a)\)[/tex]
Given the expression [tex]\(\ln\left(\frac{5 x^2 y^4}{w^3}\right)\)[/tex]:
1. Apply the logarithmic property for division, [tex]\(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)[/tex]:
[tex]\[
\ln\left(\frac{5 x^2 y^4}{w^3}\right) = \ln\left(5 x^2 y^4\right) - \ln(w^3)
\][/tex]
2. Apply the logarithmic property for multiplication, [tex]\(\ln(ab) = \ln(a) + \ln(b)\)[/tex], to [tex]\(\ln\left(5 x^2 y^4\right)\)[/tex]:
[tex]\[
\ln\left(5 x^2 y^4\right) = \ln(5) + \ln(x^2) + \ln(y^4)
\][/tex]
3. Apply the logarithmic property for exponentiation, [tex]\(\ln(a^n) = n \cdot \ln(a)\)[/tex]:
[tex]\[
\ln(x^2) = 2 \cdot \ln(x) \quad \text{and} \quad \ln(y^4) = 4 \cdot \ln(y)
\][/tex]
Thus,
[tex]\[
\ln\left(5 x^2 y^4\right) = \ln(5) + 2 \cdot \ln(x) + 4 \cdot \ln(y)
\][/tex]
4. Apply the logarithmic property for exponentiation to [tex]\(\ln(w^3)\)[/tex]:
[tex]\[
\ln(w^3) = 3 \cdot \ln(w)
\][/tex]
5. Combine everything:
[tex]\[
\ln\left(\frac{5 x^2 y^4}{w^3}\right) = \ln(5) + 2 \cdot \ln(x) + 4 \cdot \ln(y) - 3 \cdot \ln(w)
\][/tex]
Therefore, the expanded expression is:
[tex]\[
\ln(5) + 2 \cdot \ln(x) + 4 \cdot \ln(y) - 3 \cdot \ln(w)
\][/tex]
Which corresponds to the first answer option:
[tex]\[
\ln (5)+2 \cdot \ln (x)+4 \cdot \ln (y)-3 \cdot \ln (w)
\][/tex]
