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

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Transcript of Mrs. McConaughyHonors Algebra 21 Natural Logarithms Objective: To use natural logarithms.
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 (3x) a.b. g(x) = ln (x3)2
Mrs. McConaughy Honors Algebra 2 5
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
Mrs. McConaughy Honors Algebra 2 6
GRAPHING THE NATURAL LOG FUNCTION
EXAMPLE 1 Graphing the Natural Logarithmic Function
y = ex y = lnx
Mrs. McConaughy Honors Algebra 2 7
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
Mrs. McConaughy Honors Algebra 2 9
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(x2)
Note: Compare This graph to ln x before graphing.
Mrs. McConaughy Honors Algebra 2 11
EXAMPLE 5 USING THE CHANGEOFBASE 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