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Pre-CalculusChapter 3

Exponential and Logarithmic Functions

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3.2 Logarithmic FunctionsObjectives: Recognize and evaluate logarithmic

functions with base a. Graph logarithmic functions. Recognize, evaluate, and graph natural

logarithmic functions. Use logarithmic functions to model and

solve real-life problems.

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What is a logarithm?

“Logarithm” comes from two Greek

words

“Logos” = ratio

“Arithmos” = number

So, a logarithm is an exponent.

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How Were Logarithms Developed? Imagine a time before calculators…. In the 16th and 17th centuries, scientists

made significant gains in the study of astronomy, navigation, and other areas. They wanted an easier (and less error-prone) way of performing calculations, esp. multiplication and division.

Two mathematicians, John Napier and Henry Briggs, are credited with developing logarithms to simplify calculations.

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How Were Logarithms Used? Recall the property of exponents: ax · ay = a(x + y)

Napier used this property to convert complicated, difficult multiplication problems into easier, less error-prone, addition problems.

For example, 4,971.26 x 0.2459 =

10m x 10n = 10(m + n)

where 3 < m < 4 and –1 < n < 0 Briggs spent a great deal of his life

identifying values of y such that 10y = x. The value y is the logarithm. That is, y = log10 x. This data was organized into logarithmic tables.

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A Sample Table

Log.Exponent

Form Number

0 100 1 log101 = 0

0.08720 100.08720 1.222 log101.222 = 0.087

0.39076 100.39076 2.459 log102.459 = 0.39076

0.69644 100.69644 4.971 log104.971 = 0.69644

1 101 10 log1010 = 1

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Back to the Example Original statement: 4,971.26 x 0.2459 Rewrite in scientific notation:

4.97126 x 103 x 2.459 x 10-1

Use values from table:100.69644 x 103 x 100.39076 x 10-1

Simplify: 103.08720

103 x 100.08720

Use values from table:1, 000 x 1.222 = 1,222

…Fortunately, we have calculators!

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Definition of Logarithmic Function The logarithmic function with base a is

given by

f (x) = loga x

where x > 0, a > 0, and a ≠ 1.

That is,y = loga x if and only if x = a y

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Example 1 Rewrite each as an exponential expression

and solve.

a. f (x) = log2 32

b. f (x) = log3 1

c. f (x) = log4 2

d. f (x) = log10 1/100

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Example 2 Use your calculator to solve f (x) = log10 x at

each value of x.

a. x = 10

b. x = 2.5

c. x = –2

d. x = ¼

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Properties of Logarithms

1. loga 1 = 0 because a0 = 1.

2. loga a = 1 because a1 = a.

3. loga ax = x and a loga x = x Inverse Properties

4. If loga x = loga y, then x = y. One-to-One

Property

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Example 3

a. Solve for x: log2 x = log2 3

b. Solve for x: log4 4 = x

c. Simplify: log5 5x

d. Simplify: 7 log7 14

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Graph of Logarithmic Function To sketch the graph of y =

loga x, first graph y = ax, then graph its inverse.

Example: Graph f (x) = log2 x.

a. Construct a table of values for g(x) = 2x and plot the points.

b. The function f (x) = log2 x is the inverse of g(x) = 2x.

Note: Inverse functions are reflections in the line y = x.

                     

                     

                     

                     

                     

                     

                     

                     

                     

                     

                     

                     

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Logarithmic Function The logarithmic function

is the inverse of the exponential function.

Many real-life phenomena with a slow rate of growth can be modeled by logarithmic functions.

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Characteristics of the Logarithmic Function

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Transformations of f (x) = loga x

The graph of g(x) = loga (x ± h) is a _________ shift

of f.

The graph of h(x) = loga (x) ± k is a _________ shift

of f.

The graph of j(x) = – loga (x) is a reflection of f

______ .

The graph of k(x) = loga (– x) is a reflection of f

______ .

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Example 4 Each of the following

functions is a transformation of the graph of f (x) = log10 x. Describe the transformation and graph.

a. g(x) = log10 (x – 1)

b. h (x) = 2 + log10 x

                     

                     

                     

                     

                     

                     

                     

                     

                     

                     

                     

                     

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Natural Logarithmic Function

For x > 0,y = ln x if and only if x = e y

The function given by

f (x) = loge x = ln x

is called the natural logarithmic

function.

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Example 5 Use your calculator to solve f (x) = ln x at each

value of x.

a. x = 2

b. x = 0.3

c. x = –1

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Properties of Natural Logs

1. ln 1 = 0 because e0 = 1.

2. ln e = 1 because e1 = e.

3. ln ex = x and e ln x = x. Inverse

Properties

4. If ln x = ln y, then x = y. One-to-One

Property

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Example 6 Use properties of natural logs to

rewrite each expresssion.

5ln.2

1ln.1

e

e

e

e

ln2.4

ln.3 0

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Graph of Natural Log Function The graph of y = ln x is the reflection of the

graph y = ex in the line y = x.

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Domain of Log Functions Find the domain of each function.

a. f (x) = ln (x – 2)

b. g(x) = ln (2 – x)

c. h(x) = ln x2

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Transformations of f (x) = ln x

The graph of g(x) = ln (x ± h) is a ___________

shift of f.

The graph of h(x) = ln (x) ± k is a ___________

shift of f.

The graph of j(x) = – ln (x) is a reflection of f

________.

The graph of k(x) = ln (– x) is a reflection of f

________.

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Example 7 Students participating in a psychology experiment

attended several lectures on a subject and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model

f (t) = 75 – 6 ln (t + 1), 0 ≤ t ≤ 12

where t is the time in months.a. What was the average score on the original (t = 0)

exam?b.What was the average score at the end of t = 2 months?c. What was the average score at the end of t = 6 months?

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Homework 3.2 Worksheet 3.2