12.3 Properties of Logarithms

Size: px

Start display at page:

Download "12.3 Properties of Logarithms"

Melanie Wade
5 years ago
Views:

1 12.3 Properties of Logarithms The Product Rule Let b, and N be positive real numbers with b 1. N = + N The logarithm of a product is the sum of the logarithms of the factors. Eample 1: Use the product rule to epand each logarithmic epression. Assume all variables and variable epressions represent positive numbers. a. log 4 7 b. log 4 ( 7( 2) ) c. log( 10) The Quotient Rule Let b, and N be positive real numbers with b 1. N = N The logarithm of a quotient is the difference of the logarithms. Eample 2: Use the quotient rule to epand each logarithmic epression. Assume all variables and variable epressions represent positive numbers. a.. log 2 8 b. log c. ln 8.7 e 5 Note: Portions of this document are ecerpted from the tetbook Introductory and Intermediate

2 The Power Rule Let b and be positive real numbers with b 1, and let p be any real number. p = p The logarithm of a number with an eponent is the product of the eponent and the logarithm of that number. Eample 3: Use the power rule to epand each logarithmic epression. Assume all variables and variable epressions represent positive numbers. a. log 3 6 b. log 2 ( 7) 4 c. log 6 Epanding Logarithmic Epressions Summary of Properties for Epanding Logarithmic Epressions 1. N = + N Product Rule 2. N = log log N b b Quotient Rule 3. p = p Power Rule Epanding logarithmic epressions may require that you use more than one property. Eample 4: Use logarithmic properties to epand each epression as much as possible. Assume all variables and variable epressions represent positive numbers. a. log Note: Portions of this document are ecerpted from the tetbook Introductory and Intermediate

3 b. log c. 4 3 y d. log 4 25y 3 Condensing Logarithmic Epressions To condense a logarithmic epression, we write a sum or difference of two logarithmic epressions as a single logarithmic epression. Use the properties of logarithms to do so. Restatement of Properties of Logarithms: 1. + N = N Product Rule 2. N = Quotient Rule N 3. p = p Power Rule Eample 5: Write as a single logarithm. Assume all variables and variable epressions represent positive numbers. a. log25 + log4 b. log2 + log4 c. log( 1) + log( + 4) Note: Portions of this document are ecerpted from the tetbook Introductory and Intermediate

4 d. 2log log4 e. 7log 4 5 log 4 8 f. 1 2 log 2 + 2log 2 5y 2 The Change-of-Base Property For any logarithmic bases a and b, and any positive number, = log a log a b The logarithm of with base b is equal to the logarithm of with any new base divided by the logarithm of b with that new base. Since calculators generally have keys for only common or natural logs, the change of base formula must be used to evaluate logarithms with bases other than 10 or e. If the new base, a, is chosen to be 10 or e, the change-of-base formula becomes: = log logb or = ln lnb Eample 6: Evaluate each logarithm. Round your answer to the nearest hundredth. a. log b. log c. log Note: Portions of this document are ecerpted from the tetbook Introductory and Intermediate

5 Eample 7: Use the change-of-base formula and your graphing calculator to graph f = log 2 ( 1). Indicate any vertical asymptotes with a dotted line. = 2log and = log 2. Show the graphs on the grid below Eplain why the Eample 8: Use your graphing calculator to graph f f graphs are different. Note: Portions of this document are ecerpted from the tetbook Introductory and Intermediate

6 Answers Section 12.3 Eample 1: a. log log 4 b. log log 4 + log 4 2 c. 1+ log Eample 2: a. 3 log 2 b. 2 log5 c. ln8.7 5 Eample 3: a. 6log 3 b. 4log 2 7 c. 1 2 log 6 Eample 4: a. 1+ 2log 2 b log c y d. 1 2 log 4 log log 4 y Note: Portions of this document are ecerpted from the tetbook Introductory and Intermediate

7 Eample 5: a. 2 b. log 8 [( + 4) ] c. log 1 d. log e. log f. log 2 25y 4 Eample 6: a b c Eample 7: Note: Portions of this document are ecerpted from the tetbook Introductory and Intermediate

8 Eample 8: The domain of f() = log 2 2 is the set of all real numbers ecept 0 but the domain of f() = 2log 2 is the {/ >0}. f() = log 2 2 f() = 2log 2 Note: Portions of this document are ecerpted from the tetbook Introductory and Intermediate

Name. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Name. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. REVIEW Eam #3 : 3.2-3.6, 4.1-4.5, 5.1 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the Leading Coefficient Test to determine the end behavior