1. **Problem:** Use the properties of logarithms to expand the expression $3\ln a - \ln b + 5 \ln c$. 2. **Formula and rules:** - The logarithm of a product: $\ln(xy) = \ln x + \ln y$ - The logarithm of a quotient: $\ln\left(\frac{x}{y}\right) = \ln x - \ln y$ - The logarithm of a power: $\ln(x^k) = k \ln x$ 3. **Apply the properties:** The expression is already expanded, but we can write it as a single logarithm using the reverse properties: $$3\ln a - \ln b + 5 \ln c = \ln(a^3) - \ln b + \ln(c^5)$$ 4. **Combine the logarithms:** $$= \ln\left(\frac{a^3 c^5}{b}\right)$$ 5. **Explanation:** We used the power rule to rewrite coefficients as exponents inside the logarithms. Then, subtraction of logarithms corresponds to division inside the log. Addition corresponds to multiplication inside the log. **Final answer:** $$3\ln a - \ln b + 5 \ln c = \ln\left(\frac{a^3 c^5}{b}\right)$$