1. Mrs. McConaughyHonors Algebra 21 Natural Logarithms Objective: To use natural logarithms

2. Mrs. McConaughyHonors Algebra 22 The logarithmic function will help you to understand diverse phenomena including earthquake intensity, human memory, and the pace of life in large cities. California Earthquake, Oct. 1989

3. Mrs. McConaughyHonors Algebra 23 VOCABULARY The logarithmic function with base e is called the natural logarithmic function. The Natural Logarithm If x is a positive real number, then the natural logarithm of x is denoted by _____________________. NOTE: The second notation is more common. A function given by f(x) = ln (x + c) is called a natural logarithmic function. Like the domain of all logarithmic functions, the domain of ln x is ______________________; the domain of ______________________. the set of all positive real numbers ln (x + c) is x: x + c > 0 log e x = ln x.

4. Mrs. McConaughyHonors Algebra 24 CHECK POINT Find the domain of each function. a.f(x) = ln (3-x) a.b. g(x) = ln (x-3) 2

5. Mrs. McConaughyHonors Algebra 25 Evaluating Functions of the Form f(x) = ln x Most scientific calculators have a special key for evaluating natural logarithms. For example, to evaluate ln 2 on the TI- calculators, you can use the key strokes: 2 : _______ lnenter 0.6931471806

6. Mrs. McConaughyHonors Algebra 26 GRAPHING THE NATURAL LOG FUNCTION EXAMPLE 1 Graphing the Natural Logarithmic Function y = e x y = lnx

7. Mrs. McConaughyHonors Algebra 27 SIMPLIFYING NATURAL LOG EXPRESSIONS The basic properties of logarithms can be applied to natural logs. Recall: log e x = ln x Properties of Natural Logarithms General PropertiesNatural Logarithms 1. log b 1 = _____1. ln1 = ___ because ____________. 2. log b b = _____2. ln e = ___ because ____________. 3. log b b x = _____3. ln e x = ___ because ___________. 4. b log b x = x4. e ln x = x 0 1 x 0 e 0 = 1 1 e 1 = e x e x = e x

8. Mrs. McConaughyHonors Algebra 28 EVALUATING NATURAL LOGS EXAMPLE 2Using Properties of Logarithms to Evaluate Natural Logarithms NOTE: The property of ln e x = x can be used to evaluate natural logs involving powers of e. ln e 2 = ____ ln e 3 = ____ ln e 7.1 = ____ ln 1/e = ____ 2 37.1

9. Mrs. McConaughyHonors Algebra 29 EXAMPLE 3 Expanding and Condensing Natural Logarithms a. ln 3x = _________________ b. ln x 3 y = _______________ c.ln x – ln 2 = ______________ ln 3 + ln x 3ln x + ln y ln x/2

10. Mrs. McConaughyHonors Algebra 210 EXAMPLE 4GRAPHING THE NATURAL LOG FUNCTION Graph: f (x) = 3 – ln(x-2) Note: Compare This graph to ln x before graphing.

11. Mrs. McConaughyHonors Algebra 211 EXAMPLE 5 USING THE CHANGE-OF-BASE FORMULA You can use change of base formula for evaluating natural logarithms: Use a calculator to evaluate log 3 12:_____ Check your answer: _________________ log a x = ln x ÷ ln a 2.262

12. Mrs. McConaughyHonors Algebra 212 Final Checks for Understanding 1.Explain why ln e = 1. 2.Explain why ln e 6 = 6 3.Sketch the graph of g(x) = - ln(x). What is the domain of the function? What is the range? How is the graph related to the graph of f(x) = lnx? 4.Explain how to use natural logarithms to evaluate log 6 10.

13. Mrs. McConaughyHonors Algebra 213 Homework Assignment: Natural Logs WS