In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log7 (7/x)

Key Concepts

Properties of Logarithms

The properties of logarithms are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log_b(mn) = log_b(m) + log_b(n)), the quotient rule (log_b(m/n) = log_b(m) - log_b(n)), and the power rule (log_b(m^k) = k * log_b(m)). Understanding these properties is essential for expanding and simplifying logarithmic expressions.

Quotient Rule of Logarithms

The quotient rule states that the logarithm of a quotient is the difference of the logarithms of the numerator and denominator. Specifically, log_b(m/n) = log_b(m) - log_b(n). This rule is particularly useful for breaking down complex logarithmic expressions into simpler components, making it easier to evaluate or expand them.

Change of Base Formula

The change of base formula allows you to convert logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This is useful when dealing with logarithmic expressions that may not be easily evaluated in their original base, providing flexibility in calculations and simplifications.