Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 10 Exponential and Logarithmic Functions

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 10.5 Properties of Logarithms

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Properties of Logarithms This section examines several properties of logarithms that allow you to simplify expressions. Recall that a logarithm is an exponent, so logarithmic properties are restatements of exponential properties.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Note that although the property allows you to split the logarithm of a product into the sum of two logarithms, we usually use the property to consolidate a sum of logarithms into a single logarithm. Product Property of Logarithms If x, y, and b are positive real numbers and b ≠ 1, then log b xy = log b x + log b y

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall a. log 7 x + log 7 4 = log 7 (4 x) Example Write each sum as a single logarithm. a. log 7 x + log 7 4 b. log z 5 + log z 8 Solution b. log z 5 + log z 8 = log z (5 8) = log 7 4x= log z 40

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall log 2 x + log log 2 (x + 3) log 2 10x(x + 3) Example Write as a single logarithm. log 2 x + log log 2 (x + 3) Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Note that although the property allows you to split the logarithm of a quotient into the difference of two logarithms, we usually use the property to consolidate a difference of logarithms into a single logarithm. Quotient Property of Logarithms If x, y, and b are positive real numbers and b ≠ 1, then

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Write each difference as a single logarithm. a. log 7 x – log 7 4 b. log z 40 – log z 8 = log z 5 Example b. log z 40 – log z 8 Solution a. log 7 x – log 7 4

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Write the difference as a single logarithm. log 4 x – log 4 10 – log 4 (x + 3) Example Solution log 4 x – log 4 10 – log 4 (x + 3) Apply the quotient property.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Power Property of Logarithms If x and b are positive real numbers, b ≠ 1, and r is a real number, then log b x r = r log b x

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the power property to rewrite each expression. a. log 7 x – 3 b. Example a. log 7 x – 3 = – 3 log 7 x Solution b.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Write the expression as a single logarithm. 2 log 5 x + log 5 4 – 3 log 5 (x + 5) Example 2 log 5 x + log 5 4 – 3 log 5 (x + 5) Solution = log 5 x 2 + log 5 4 – log 5 (x + 5) 3

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Write the expression as a sum or difference of multiples of logarithms. Example Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall If log b 2 = 0.43 and log b 3 = 0.68, use the properties of logarithms to evaluate. a. log b 16 Example = log b 2 4 Solution a. log b 16 = 4 log b 2 = 4(0.43) = 1.72

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Properties of Logarithm If x, y, and b are positive real numbers, b ≠ 1, and r is a real number, then