3.2 logarithmic functions

3.2 Logarithms

• All of the material in this slideshow is important, however the only slides than must be included in your notes are the Recap slides at the end.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1


IntroSolving for an

answerSolving for a base Solving for an

exponent

1. 34 = x

81 = x

2. x2 = 49

x = ±7

3. 3x = 729

log3729=3New Calc:

Alpha window

5:logBASE(

Older Calc:

log(729)/log(3)

Answer: 6

b/c 36 = 729


Now you try…

4. 25 = x 5. x3 = 125 6. 5x = 3125

Plug in calc 32 = x

3 3 3 125

5

x

x

5log 3125

log expbase

x

ans


Every logarithmic equation has an equivalent exponential form:

y = loga x is equivalent to x = a y

A logarithmic function is the inverse function of an exponential function.

A logarithm is an exponent!


y = log2( )2

1 = 2 y

2

1

Examples: Write the equivalent exponential equation and solve for y.

1 = 5 yy = log51

16 = 4y y = log416

16 = 2yy = log216

SolutionEquivalent Exponential

Equation

Logarithmic Equation

16 = 24 y = 4

2

1= 2-1 y = –1

16 = 42 y = 2

1 = 50 y = 0


3.2 Material • Logarithms are used when solving for an exponent.

baseexp =ans logbaseans=exp

• In your calculator: log is base 10 ln is base e

Properties of Logarithms and Ln

1. loga 1 = 0

2. logaa = 1

3. logaax = x

• The graph is the inverse of y=ax (reflected over y =x )

VA: x = 0 , x-intercept (1,0), increasing


The graphs of logarithmic functions are similar for different values of a. f(x) = loga x

x-intercept (1, 0)

increasing

reflection of y = a x in y = x

VA: x = 0

Graph of f (x) = loga x

x

y y = x

y = log2 x

y = a xy-axis

verticalasymptote

x-intercept(1, 0)


Example: Graph the common logarithm function f(x) = log10 x.

y

x5

–5

f(x) = log10 x

x = 0 vertical

asymptote

(1, 0) x-intercept


The function defined by f(x) = loge x = ln x

is called the natural logarithm function.

Use a calculator to evaluate: ln 3, ln –2, ln 100

ln 3ln –2ln 100

Function Value Keystrokes Display

LN 3 ENTER 1.0986122ERRORLN –2 ENTER

LN 100 ENTER 4.6051701

y = ln xy

x5

–5

y = ln x is equivalent to e y = x


Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x4. If ln x = ln y, then x = y. one-to-one property

Examples: Simplify each expression.

2

1ln

e 2ln 2 e inverse property

eln3 3)1(3 property 2

00 1ln property 1


Example: The formula (t in years) is used to estimate

the age of organic material. The ratio of carbon 14 to carbon 12

in a piece of charcoal found at an archaeological dig is .

How old is it?

82231210

1 t

eR

To the nearest thousand years the charcoal is 57,000 years old.

original equation158223

12 10

1

10

1

t

e

multiply both sides by 10123

8223

10

1

t

e

take the ln of both sides1000

1lnln 8223

t

e

inverse property1000

1ln

8223

t

56796907.6 82231000

1ln 8223

t

1510

1R


3.2 Material Recap • Logarithms are used when solving for an exponent.

logab=c ac =b

• In your calculator: log is base 10 ln is base e

Properties of Logarithms and Ln

1. loga 1 = 0 because when written in exponential form a0 = 1

2. logaa = 1 because when written in exponential form a1 = a

3. logaax = x because when written in exponential form ax = ax


Graph the function f(x) = log x.

y

x5

–5

f(x) = log10 x

x = 0 vertical

asymptote

(1, 0) x-intercept

• The graph is the inverse of y=ax (reflected over y =x )

VA: x = 0 , x-intercept (1,0), increasing

3.2 Material Recap


Practice ProblemsPage 236 (all work and solutions on blackboard)

•#1–16 pick any 4

•17–22 pick any 3

•27–29 all




•69–71 all


3.2 logarithmic functions

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base

logarithms

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