Logarithms: Introduction and Properties

A logarithm answers the question: "What exponent gives me this number?" [formula] Examples: - log_2(8) = 3 because 2^3 = 8 - log_{10}(1000) = 3 because 10^3 = 1000 - log_5(25) = 2 because 5^2 = 25 Special cases: - log_b(1) = 0 (because b^0 = 1) - log_b(b) = 1 (because b^1 = b) - ln(x) = log_e(x) (natural log, e approx 2.718)

Example: Evaluate log_3(81)

Product rule: log_b(xy) = log_b(x) + log_b(y) Quotient rule: log_bleft(frac{x}{y}right) = log_b(x) - log_b(y) Power rule: log_b(x^n) = n cdot log_b(x) Change of base: log_b(x) = frac{log(x)}{log(b)} = frac{ln(x)}{ln(b)}

Example: Simplify log_2(8) + log_2(4)