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### Transcript of Mrs. McConaughyHonors Algebra 21 Natural Logarithms Objective: To use natural logarithms

• Mrs. McConaughyHonors Algebra 2*Natural LogarithmsObjective: To use natural logarithms

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*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

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*VOCABULARYThe logarithmic function with base e is called the natural logarithmic function. the set of all positive real numbersln (x + c) is x: x + c > 0log e x = ln x.

The Natural LogarithmIf 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 ______________________.

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*CHECK POINTFind the domain of each function.

f(x) = ln (3-x) b. g(x) = ln (x-3)2

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*Evaluating Functions of the Form f(x) = ln xlnenter0.6931471806

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 : _______

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*GRAPHING THE NATURAL LOG FUNCTION y = exy = lnx

EXAMPLE 1Graphing the Natural Logarithmic Function

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*SIMPLIFYING NATURAL LOG EXPRESSIONS

The basic properties of logarithms can be applied to natural logs.Recall: log e x = ln x

01x0e 0 = 11e 1= exe x= ex

Properties of Natural LogarithmsGeneral PropertiesNatural Logarithms1. log b 1 = _____1. ln1 = ___ because ____________.2. log b b = _____2. ln e = ___ because ____________.3. log b bx = _____3. ln ex = ___ because ___________.4. b log bx = x4. e ln x = x

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*EVALUATING NATURAL LOGS237.1-1

EXAMPLE 2Using Properties of Logarithms to Evaluate Natural LogarithmsNOTE: The property of ln ex = 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 = ____

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*ln 3 + ln x3ln x + ln yln x/2

EXAMPLE 3Expanding and Condensing Natural Logarithmsa. ln 3x = _________________

b. ln x 3 y = _______________

ln x ln 2 = ______________

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*

EXAMPLE 4GRAPHING THE NATURAL LOG FUNCTIONGraph: f (x) = 3 ln(x-2)

Note: Compare This graph to ln x before graphing.

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*log a x = ln x ln a2.262

EXAMPLE 5USING THE CHANGE-OF-BASE FORMULAYou can use change of base formula for evaluating natural logarithms:

Use a calculator to evaluate log 3 12:_____ Check your answer: _________________

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*Final Checks for UnderstandingExplain why ln e = 1.Explain why ln e 6 = 6Sketch 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?Explain how to use natural logarithms to evaluate log6 10.

Honors Algebra 2

• Mrs. McConaughyHonors Algebra 2*Homework Assignment: Natural Logs WS

Honors Algebra 2