Question

expanding logarithmic expressions
expand the expression.
\\(\ln \frac{2a}{b} =\\)
\\(\ln 2 + \ln a - \ln b\\)
\\(\ln 2 - \ln a - \ln b\\)
\\(2 \ln a - \ln b\\)

Explanation:

Step1: Recall logarithm quotient rule

The quotient rule for logarithms states that $\ln \frac{x}{y} = \ln x - \ln y$. So for $\ln \frac{2a}{b}$, we can first apply the quotient rule: $\ln \frac{2a}{b}=\ln(2a)-\ln b$.

Step2: Recall logarithm product rule

The product rule for logarithms states that $\ln(xy)=\ln x + \ln y$. Applying this to $\ln(2a)$, we get $\ln(2a)=\ln 2+\ln a$.

Step3: Combine the results

Substituting $\ln(2a)=\ln 2+\ln a$ into $\ln(2a)-\ln b$, we have $\ln \frac{2a}{b}=\ln 2+\ln a - \ln b$.

Answer:

$\ln 2 + \ln a - \ln b$ (corresponding to the option "ln 2 + ln a - ln b")