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Sec 4.3 Laws of Logarithms Objective: •To understand the laws of logarithms, including the change of base formula. Laws of Logarithms Let a be a positive number, with a ≠ 1. Let A, B, and C be any real numbers with A > 0 and B > 0, then the following laws apply: loga ( AB) loga A loga B •The logarithm of a product of numbers is the sum of the logarithms of the numbers. Ex. 1 Evaluate log 4 2 log 4 32 A loga loga A loga B B •The logarithm of a quotient of numbers is the difference of the logarithms of the numbers. Ex. 2 Evaluate log 2 80 log 2 5 C log C loga A a A The logarithm of a power of a number is the exponent times the logarithm of the number. Ex. 3 Evaluate 1 log 8 3 Expanding and Combining Logarithmic Expressions Ex 4. Use the Laws of Logarithms to expand each expression. (a) log2 (6 x ) 3 (b) log5 x y ab (c) ln 3 c 6 Ex 5. Combine the following into a single logarithm. 3 log x + ½ log(x + 1) Ex 6. Combing the following into a single logarithm. 3 ln s + ½ ln t – 4 ln(t2 + 1) Change of Base loga x logb x loga b Ex 7. Use the Change of Base Formula and common or natural logarithms to evaluate each logarithm, correct to five decimal places. (a) log8 5 (b) log9 20 Solving an Exponential Equation Ex 8. Solve for x. Round to 3 decimal places. x 2 3 7 Ex. 9 Solve for x. Round to 3 decimal places. 8e 20 2x Ex 10. Solve for x. Round to 3 decimal places. 3 2 x e 4 HW – Log Worksheet