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. log N^(-6)

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^p) = p * log_b(M)). Understanding these properties is essential for expanding and simplifying logarithmic expressions.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, N^(-6) can be rewritten as 1/(N^6). This concept is crucial when dealing with logarithmic expressions involving negative exponents, as it allows for the application of logarithmic properties effectively.

Logarithmic Evaluation

Evaluating logarithmic expressions involves determining the value of the logarithm based on its definition. For instance, log_b(a) answers the question: 'To what power must b be raised to obtain a?' In the context of expanding log N^(-6), understanding how to evaluate logarithms without a calculator can help in simplifying the expression further.