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5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate, and graph natural logs Use logarithmic functions to model and solve real-life problems.

5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

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Page 1: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

5.2 Logarithmic Functions & Their Graphs

Goals—Recognize and evaluate logarithmic functions with base a

Graph Logarithmic functionsRecognize, evaluate, and graph natural logs

Use logarithmic functions to model and solve real-life problems.

Page 2: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

f(x) = 3x

Is this function one to one?

Horizontal Line test?

Does it have an inverse?

Page 3: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Logarithmic function with base “a”

• Def’n of Logarithmic function with base “a”

For x > 0, a > 0, and a 1,y = logax if and only if x = ay

The function given by f(x) = logax read as “log base a of x”

is called the logarithmic function with base a.

Page 4: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Write the logarithmic equation in exponential form

log381 = 4 log168 = 3/4

Write the exponential equation in logarithmic form

82 = 64 4-3 = 1/64

34 = 81 163/4 = 8

log 8 64 = 2 log4 (1/64) = -3

Page 5: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Evaluating Logs

f(x) = log232 f(x) = log42

f(x) = log31

f(x) = log10(1/100)

Step 1- rewrite it as an exponential equation. 2y = 32

Step 2- make the bases the same.

2y = 25

Therefore, y = 5

4y = 222y = 22y = 2y = 1

3y = 1y = 0

10y = 1/10010y = 10-2

y = -2

Page 6: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Evaluating Logs on a Calculator

f(x) = log x when x = 10 f(x) = 1 when x = 1/3 f(x) = -.4771 when x = 2.5 f(x) = .3979 when x = -2 f(x) = ERROR!!! Why???

You can only use a calculator when the base is 10

Page 7: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Properties of Logarithms

• loga1 = 0 because a0 = 1

• logaa = 1 because a1 = a

• logaax = x and alogax = x

• logax = logay, then x = y

Page 8: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Simplify using the properties of logs

log41

log77

6log620

Rewrite as an exponent4y = 1 So y = 0

Rewrite as an exponent7y = 7 So y = 1

Page 9: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Use the 1-1 property to solve

log3x = log312

log3(2x + 1) = log3x

log4(x2 - 6) = log4 10

x = 12

2x + 1 = xx = -1

x2 - 6 = 10x2 = 16x = 4

Page 10: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

f(x) = 3x

Graphs of Logarithmic Functions

So, the inverse would beg(x) = log3x

Make a T chart

Domain—Range?

Asymptotes?

Page 11: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Graphs of Logarithmic Functions

g(x) = log4(x – 3)

Make a T chart

Domain—Range?

Asymptotes?

Page 12: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Graphs of Logarithmic Functions

g(x) = log5(x – 1) + 4

Make a T chart

Domain—Range?

Asymptotes?

Page 13: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Natural Logarithmic Functions

• The function defined by f(x) = loge x = ln x, x > 0

is called the natural logarithmic function.

Page 14: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Evaluatef(x) = ln x when x = 2 f(x) = .6931 when x = -1 f(x) = Error!!! Why???

Page 15: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Properties of Natural Logarithms

ln 1 = 0 because e0 = 1

ln e = 1 because e1 = e

ln ex = x and elnx = x (Think…they are inverses of each other.)

If ln x = ln y, then x = y

Page 16: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Use properties of Natural Logs to simplify each expression

ln (1/e) = ln e-1 = -1 eln 5 = 5 2 ln e = 2

Page 17: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Graphs of Natural Logs

g(x) = ln(x + 2)

Make a T chart

Domain—Range?

Asymptotes?

2 Undefined 3 4

Page 18: 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

Graphs of Natural Logsg(x) = ln(2 - x)

Make a T chart

Domain—Range?

Asymptotes?

2 Undefined 1 0