Slide Copyright © 2012 Pearson Education, Inc.

9.4 Properties of Logarithmic Functions ■ Logarithms of Products ■ Logarithms of Powers ■ Logarithms of Quotients ■ Using the Properties Together

Slide 9- 3 Copyright © 2012 Pearson Education, Inc. Logarithms of Products The first property we discuss is related to the product rule for exponents:

Slide 9- 4 Copyright © 2012 Pearson Education, Inc. The Product Rule for Logarithms For any positive numbers M, N and a (The logarithm of a product is the sum of the logarithms of the factors.)

Slide 9- 5 Copyright © 2012 Pearson Education, Inc. Example Solution that is a sum of logarithms: log 3 (9 · 27) = Express as an equivalent expression log 3 (9 · 27). log log As a check, note that log 3 (9 · 27) = log = = 243 and that log log 3 27 = = = 9 and 3 3 = 27

Slide 9- 6 Copyright © 2012 Pearson Education, Inc. Example Solution that is a single logarithm: Express as an equivalent expression = log a (42). Using the product rule for logarithms log a 6 + log a 7. log a 6 + log a 7 = log a (6 · 7)

Slide 9- 7 Copyright © 2012 Pearson Education, Inc. Logarithms of Powers The second basic property is related to the power rule for exponents:

Slide 9- 8 Copyright © 2012 Pearson Education, Inc. The Power Rule for Logarithms For any positive numbers M and a and any real number p, (The logarithm of a power of M is the exponent times the logarithm of M.)

Slide 9- 9 Copyright © 2012 Pearson Education, Inc. Example Solution expression that is a product: Use the power rule to write an equivalent = log 4 x 1/2 Using the power rule for logarithms a) log a 6 –3 ; a) log a 6 -3 = –3log a 6 = ½ log 4 x Using the power rule for logarithms

Slide Copyright © 2012 Pearson Education, Inc. Logarithms of Quotients The third property that we study is similar to the quotient rule for exponents:

Slide Copyright © 2012 Pearson Education, Inc. The Quotient Rule for Logarithms For any positive numbers M, N and a (The logarithm of a quotient is the logarithm of the dividend minus the logarithm of the divisor.)

Slide Copyright © 2012 Pearson Education, Inc. Example Solution that is a difference of logarithms: log 3 (9/y) = Express as an equivalent expression log 3 (9/y). log 3 9 – log 3 y. Using the quotient rule for logarithms

Slide Copyright © 2012 Pearson Education, Inc. Example Solution that is a single logarithm: Express as an equivalent expression log a 6 – log a 7. log a 6 – log a 7 = log a (6/7) Using the quotient rule for logarithms “in reverse”

Slide Copyright © 2012 Pearson Education, Inc. Using the Properties Together

Slide Copyright © 2012 Pearson Education, Inc. Example Solution using individual logarithms of x, y, and z. Express as an equivalent expression = log 4 x 3 – log 4 yz = 3log 4 x – log 4 yz = 3log 4 x – (log 4 y + log 4 z) = 3log 4 x –log 4 y – log 4 z

Slide Copyright © 2012 Pearson Education, Inc. Solution continued

Slide Copyright © 2012 Pearson Education, Inc. CAUTION! Because the product and quotient rules replace one term with two, it is often best to use the rules within parentheses, as in the previous example.

Slide Copyright © 2012 Pearson Education, Inc. Example Solution that is a single logarithm. Express as an equivalent expression = log b x 1/3 – log b y 2 + log b z

Slide Copyright © 2012 Pearson Education, Inc. The Logarithm of the Base to an Exponent For any base a, (The logarithm, base a, of a to an exponent is the exponent.)

Slide Copyright © 2012 Pearson Education, Inc. Example Solution Simplify: a) log b) log 3 3 –3.4 a) log =8 b) log 3 3 –3.4 = –3.4 8 is the exponent to which you raise 6 in order to get 6 8.

Slide Copyright © 2012 Pearson Education, Inc. For any positive numbers M, N, and a