Question

expand the expression.
\\(\ln \frac{4y^5}{x^2}\\)
\\(\ln 20 + \ln y - \ln 2x\\)
\\(\ln 20 + \ln y - \ln 2 - \ln x\\)
\\(\ln 4 - 2\ln x - 5\ln y\\)
\\(\ln 4 - 2\ln x + 5\ln y\\)

Explanation:

Step1: Apply quotient rule of logarithms

The quotient rule for logarithms states that $\ln\frac{a}{b}=\ln a - \ln b$. So, for $\ln\frac{4y^{5}}{x^{2}}$, we can rewrite it as $\ln(4y^{5})-\ln(x^{2})$.

Step2: Apply product rule of logarithms

The product rule for logarithms states that $\ln(ab)=\ln a+\ln b$. Applying this to $\ln(4y^{5})$, we get $\ln 4+\ln(y^{5})-\ln(x^{2})$.

Step3: Apply power rule of logarithms

The power rule for logarithms states that $\ln(a^{n}) = n\ln a$. Applying this to $\ln(y^{5})$ and $\ln(x^{2})$, we have $\ln 4 + 5\ln y- 2\ln x$.

Answer:

$\ln 4 - 2\ln x + 5\ln y$ (the option: $\boldsymbol{\ln 4 - 2\ln x + 5\ln y}$)