Mrs. McConaughy Honors Algebra 2 1

Natural Logarithms

Objective: To use natural logarithms

Mrs. McConaughy Honors Algebra 2 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

Mrs. McConaughy Honors Algebra 2 3

VOCABULARYThe logarithmic function with base e is called the natural logarithmic function.

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 ______________________.

the set of all positive real numbers

ln (x + c) is x: x + c > 0

log e x = ln x.

Mrs. McConaughy Honors Algebra 2 4

CHECK POINT

Find the domain of each function.

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

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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 : _______

ln enter

0.6931471806

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GRAPHING THE NATURAL LOG FUNCTION

EXAMPLE 1 Graphing the Natural Logarithmic Function

y = ex y = lnx

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SIMPLIFYING NATURAL LOG EXPRESSIONS

The basic properties of logarithms can be applied to natural logs.

Recall: log e x = ln x

Properties of Natural LogarithmsGeneral Properties Natural Logarithms

1. 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 b

x = x 4. e ln x = x

0

1

x

0e 0 = 1

1e 1= e

xe x= ex

Mrs. McConaughy Honors Algebra 2 8

EVALUATING NATURAL LOGS

EXAMPLE 2

Using Properties of Logarithms to Evaluate Natural Logarithms

NOTE: 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 = ____

2 3 7.1 -1

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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

Mrs. McConaughy Honors Algebra 2 10

EXAMPLE 4 GRAPHING THE NATURAL LOG FUNCTION

Graph: f (x) = 3 – ln(x-2)

Note: Compare This graph to ln x before graphing.

Mrs. McConaughy Honors Algebra 2 11

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

Mrs. McConaughy Honors Algebra 2 12

Final Checks for Understanding

1. Explain why ln e = 1.2. Explain why ln e 6 = 63. 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 log6 10.

Mrs. McConaughy Honors Algebra 2 13

Homework Assignment:

Natural Logs WS