The properties of logarithms are fundamental tools that allow us to manipulate and simplify expressions. These properties include the product, quotient, and power rules of logarithms:
- Product Rule: \( \ln a + \ln b = \ln (ab) \)
- Quotient Rule: \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \)
- Power Rule: \( a \ln b = \ln b^a \)
In this exercise, after applying the exponent rule, we apply these properties to combine the individual logarithms into a single logarithm.
Notice how the quotient rule is applied to the expression \( \ln (x+3)^{2/3} + \ln x^{1/3} - \ln (x^{2} - 1)^{1/3} \) to get a single logarithm: \( \ln \frac{(x+3)^{2/3} x^{1/3}}{(x^{2} - 1)^{1/3}} \). This skill in handling properties of logarithms allows consolidation into simpler expressions, making them easier to evaluate and interpret.