Inverse, Exponential and Logarithmic Functions Chapter 11 Inverse, Exponential and Logarithmic Functions

Properties of Logarithms 11.4 Properties of Logarithms

11.4 Properties of Logarithms Objectives Use the product rule for logarithms. Use the quotient rule for logarithms. Use the power rule for logarithms. Use the properties to write alternative forms of logarithmic expressions. Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms Product Rule for Logarithms Product Rule for Logarithms If x, y, and b are positive real numbers, where b ≠ 1, then logb xy = logb x + logb y. In words, the logarithm of a product is the sum of the logarithms of the factors. Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms Note on Solving Equations NOTE The word statement of the product rule can be restated by replacing “logarithm” with “exponent.” The rule then becomes the familiar rule for multiplying exponential expressions: The exponent of a product is equal to the sum of the exponents of the factors. Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms EXAMPLE 1 Using the Product Rule Use the product rule to rewrite each expression. Assume n > 0. (a) log4 (3 · 5) By the product rule, log4 (3 · 5) = log4 3 + log4 5. (b) log5 2 + log5 8 = log5 (2 · 8) = log5 16 (c) log7 (7n) = log7 7 + log7 n = 1 + log7 n log7 7 = 1 Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms EXAMPLE 1 Using the Product Rule Use the product rule to rewrite each expression. Assume n > 0. (d) log3 n4 = log3 (n · n · n · n) n4 = n · n · n · n = log3 n + log3 n + log3 n + log3 n Product rule = 4 log3 n Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms Quotient Rule for Logarithms Quotient Rule for Logarithms If x, y, and b are positive real numbers, where b ≠ 1, then logb = logb x – logb y. x y In words, the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms EXAMPLE 2 Using the Quotient Rule Use the quotient rule to rewrite each logarithm. (a) log5 3 4 = log5 3 – log5 4 = log3 , n > 0 7 n (b) log3 7 – log3 n (c) log4 64 9 = log4 64 – log4 9 = 3 – log4 9 log4 64 = 3 Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms Caution CAUTION There is no property of logarithms to rewrite the logarithm of a sum or difference. For example, we cannot write logb (x + y) in terms of logb x and logb y. Also, logb ≠ . x y logb x logb y Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms Power Rule for Logarithms Power Rule for Logarithms If x and b are positive real numbers, where b ≠ 1, and if r is any real number, then logb x r = r logb x. In words, the logarithm of a number to a power equals the exponent times the logarithm of the number. Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms EXAMPLE 3 Using the Power Rule Use the power rule to rewrite each logarithm. Assume b > 0, x > 0, and b ≠ 1. (a) log5 74 = 4 log5 7 (b) logb x3 = 3 logb x (c) log4 x8 3 = log4 Rewrite the radical expression with a rational exponent. x8/3 = log4 x 8 3 Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms Special Properties Special Properties If b > 0 and b ≠ 1, then and logb b x = x. b = x logb x Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms EXAMPLE 4 Using the Special Properties Find the value of each logarithmic expression (a) log3 37 log3 37 = 7 Since logb b x = x, (b) log5 625 = log5 54 = 4 3 log3 8 (c) = 8 Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms If x, y, and b are positive real numbers, where b ≠ 1, and r is any real number, then Product Rule logb xy = logb x + logb y logb = logb x – logb y x y Quotient Rule Power Rule logb x r = r logb x and logb b x = x. b = x logb x Special Properties Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms EXAMPLE 5 Writing Logarithms in Alternative Forms Use the properties of logarithms to rewrite each expression. Assume all variables represent positive real numbers. (a) log5 5x7 = log5 5 + log5 x7 Product rule = 1 + 7 log5 x log5 5 = 1; power rule (b) log4 a 3 c = log4 1/3 a c = log4 a c 1 3 Power rule = ( log4 a 1 3 – log4 c ) Quotient rule Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms EXAMPLE 5 Writing Logarithms in Alternative Forms Use the properties of logarithms to rewrite each expression. Assume all variables represent positive real numbers. (c) log3 mn w2 = log3 w2 – log3 mn Quotient rule = 2 log3 w – log3 mn Power rule = 2 log3 w – ( log3 m + log3 n ) Product rule = 2 log3 w – log3 m – log3 n Distributive property Notice the careful use of parentheses in the third step. Since we are subtracting the logarithm of a product and rewriting it as a sum of two terms, we must place parentheses around the sum. Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms EXAMPLE 5 Writing Logarithms in Alternative Forms Use the properties of logarithms to rewrite each expression. Assume all variables represent positive real numbers. (d) logb (h – 4) + logb (h + 3) – logb h 4 5 Power rule = logb (h – 4) + logb (h + 3) – logb h4/5 h4/5 logb = (h – 4)(h + 3) Product and quotient rules h4/5 logb = h2 – h – 12 Copyright © 2010 Pearson Education, Inc. All rights reserved.

11.4 Properties of Logarithms EXAMPLE 5 Writing Logarithms in Alternative Forms Use the properties of logarithms to rewrite each expression. Assume all variables represent positive real numbers. (e) logb (3p – 4q) logb (3p – 4q) cannot be rewritten using the properties of logarithms. There is no property of logarithms to rewrite the logarithm of a difference. Copyright © 2010 Pearson Education, Inc. All rights reserved.