Copyright © 2010 Pearson Education, Inc. All rights reserved Sec

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec Inverse, Exponential and Logarithmic Functions Chapter 11

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec Properties of Logarithms

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Properties of Logarithms Objectives 1.Use the product rule for logarithms. 2.Use the quotient rule for logarithms. 3.Use the power rule for logarithms. 4.Use the properties to write alternative forms of logarithmic expressions.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Properties of Logarithms Product Rule for Logarithms If x, y, and b are positive real numbers, where b ≠ 1, then log b xy = log b x + log b 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. Sec 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. Sec (a) log 4 (3 · 5) By the product rule, log 4 (3 · 5) = log log 4 5. Use the product rule to rewrite each expression. Assume n > 0. EXAMPLE 1 Using the Product Rule 11.4 Properties of Logarithms (b) log log 5 8= log 5 (2 · 8)= log 5 16 (c) log 7 (7 n ) = log log 7 n = 1 + log 7 n log 7 7 = 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec (d) log 3 n 4 = log 3 ( n · n · n · n ) n 4 = n · n · n · n = log 3 n + log 3 n + log 3 n + log 3 n Product rule = 4 log 3 n Use the product rule to rewrite each expression. Assume n > 0. EXAMPLE 1 Using the Product Rule 11.4 Properties of Logarithms

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Use the quotient rule to rewrite each logarithm. EXAMPLE 2 Using the Quotient Rule 11.4 Properties of Logarithms (a)log = log 5 3 – log 5 4 = log 3, n > 0 7 n (b)log 3 7 – log 3 n (c)log = log 4 64 – log 4 9 = 3 – log 4 9 log 4 64 = 3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 log b ( x + y ) in terms of log b x and log b y. Also, log b ≠. x y log b x log b y

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Use the power rule to rewrite each logarithm. Assume b > 0, x > 0, and b ≠ 1. EXAMPLE 3 Using the Power Rule 11.4 Properties of Logarithms (a)log = 4 log 5 7 = 3 log b x (b)log b x 3 (c)log 4 x8x8 3 = log 4 Rewrite the radical expression with a rational exponent. x 8/3 = log 4 x 8 3

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Find the value of each logarithmic expression EXAMPLE 4 Using the Special Properties 11.4 Properties of Logarithms (a)log log = 7Since log b b x = x, (b)log 5 625= log log 3 8 (c) = 8 = 4

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Use the properties of logarithms to rewrite each expression. Assume all variables represent positive real numbers. EXAMPLE 5 Writing Logarithms in Alternative Forms 11.4 Properties of Logarithms (a)log 5 5 x 7 = log log 5 x 7 Product rule = log 5 x log 5 5 = 1; power rule (b)log 4 a 3 c = 1/3 a c =log 4 a c 1 3 =( log 4 a 1 3 – log 4 c ) Quotient rule Power rule

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Use the properties of logarithms to rewrite each expression. Assume all variables represent positive real numbers. EXAMPLE 5 Writing Logarithms in Alternative Forms 11.4 Properties of Logarithms = log 3 w 2 – log 3 mn = 2 log 3 w – log 3 mn (c)log 3 mn w2w2 Power rule Quotient rule = 2 log 3 w – ( log 3 m + log 3 n )Product rule = 2 log 3 w – log 3 m – log 3 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. Sec Use the properties of logarithms to rewrite each expression. Assume all variables represent positive real numbers. EXAMPLE 5 Writing Logarithms in Alternative Forms 11.4 Properties of Logarithms (d)log b ( h – 4) + log b ( h + 3) – log b h 4 5 Power rule=log b ( h – 4) + log b ( h + 3) – log b h 4/5 Product and quotient rules h 4/5 log b = ( h – 4)( h + 3) h 4/5 log b = h 2 – h – 12

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