In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator.(1/2)ln x + ln y

The properties of logarithms include rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log_a(b) + log_a(c) = log_a(bc)), the quotient rule (log_a(b) - log_a(c) = log_a(b/c)), and the power rule (k * log_a(b) = log_a(b^k)). Understanding these properties is essential for condensing logarithmic expressions.

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is commonly used in calculus and exponential growth models. Recognizing that ln(x) can be manipulated using logarithmic properties is crucial for solving problems involving natural logarithms.

Condensing logarithmic expressions involves combining multiple logarithms into a single logarithm. This process often requires applying the properties of logarithms to eliminate coefficients and combine terms. For example, in the expression (1/2)ln x + ln y, one would use the power rule to rewrite (1/2)ln x as ln(x^(1/2)) and then apply the product rule to combine it with ln y.