LOGARITHMSLOGARITHMSSection 4.2Section 4.2

JMerrill, 2005JMerrill, 2005

Revised 2008Revised 2008

Exponential FunctionsExponential Functions

1. Graph the 1. Graph the exponential equation exponential equation f(x) = 2f(x) = 2xx on the graph on the graph and record some and record some ordered pairs.ordered pairs.

xx f(x)f(x)

00 11

11 22

22 44

33 88

ReviewReview

2. Is this a function?2. Is this a function?– Yes, it passes the Yes, it passes the

vertical line test (which vertical line test (which means that no x’s are means that no x’s are repeated)repeated)

3. Domain? 3. Domain?

Range?Range?

,

0,

ReviewReview

2. Is the function one-2. Is the function one-to-one? Does it have to-one? Does it have an inverse that is a an inverse that is a function?function?– Yes, it passes the Yes, it passes the

horizontal line test.horizontal line test.

InversesInverses

To graph an inverse, simply switch the x’s and To graph an inverse, simply switch the x’s and y’s (remember???)y’s (remember???)

f(x) = f(x) = f f -1-1(x) = (x) = xx f(x)f(x)

00 11

11 22

22 44

33 88

xx f(x)f(x)

11 00

22 11

44 22

88 33

Now graphNow graph

f(x)f(x) ff-1-1(x)(x)

How are the Domain and Range of How are the Domain and Range of f(x) and f f(x) and f -1-1(x) related?(x) related?

The domain of the original function is the The domain of the original function is the same as the range of the new function and same as the range of the new function and vice versa.vice versa.

f(x) = f(x) = f f -1-1(x) = (x) = xx f(x)f(x)

00 11

11 22

22 44

33 88

xx f(x)f(x)

11 00

22 11

44 22

88 33

Graphing Both on the Same GraphGraphing Both on the Same Graph

Can you tell that theCan you tell that the

functions are inversesfunctions are inverses

of each other? How?of each other? How?

Graphing Both on the Same GraphGraphing Both on the Same Graph

Can you tell that theCan you tell that the

functions are inversesfunctions are inverses

of each other? How?of each other? How?

They are symmetricThey are symmetric

about the line y = x!about the line y = x!

Logarithms and ExponentialsLogarithms and Exponentials

The inverse function of the exponential The inverse function of the exponential function with base function with base bb is called the logarithmic is called the logarithmic function with base function with base bb. .

Definition of the Logarithmic Definition of the Logarithmic FunctionFunction

For x > 0, and b > 0, b For x > 0, and b > 0, b 1 1 y = logy = logbbx iff bx iff by y = x= x

The equation y = logThe equation y = logbbx and bx and by y = x are = x are

different ways of expressing the same thing. different ways of expressing the same thing. The first equation is the logarithmic form; The first equation is the logarithmic form; the second is the exponential form.the second is the exponential form.

Location of Base and ExponentLocation of Base and Exponent

Logarithmic: logLogarithmic: logbbx = yx = y

Exponential: bExponential: byy = x = x

Exponent

Base

Exponent

BaseThe 1st to the last = the middle

Changing from Logarithmic to Changing from Logarithmic to Exponential FormExponential Form

a.a. loglog55 x = 2 x = 2 meansmeans 5522 = x = x So, x = 25So, x = 25

b.b. loglogbb64 = 364 = 3meansmeans bb33 = 64 = 64 So, b = 4 since 4So, b = 4 since 433 = 64 = 64

You do:You do: c. logc. log2216 = x16 = x meansmeans

So, x = 4 since 2So, x = 4 since 244 = 16 = 16

d. logd. log25255 = x 5 = x meansmeans So, x = ½ since the square root of 25 = 5!So, x = ½ since the square root of 25 = 5!

22xx = 16 = 16

2525xx = 5 = 5

Changing from Exponential to Changing from Exponential to LogarithmicLogarithmic

a.a. 121222 = x = x meansmeans log log1212x = 2x = 2

b.b. bb3 3 = 9= 9 meansmeans log logbb9 = 39 = 3

You do:You do: c. cc. c44 = 16 = 16 meansmeans d. 7d. 722 = x = x meansmeans

loglogcc16 = 416 = 4

loglog77x = 2x = 2

Properties of LogarithmsProperties of Logarithms

Basic Logarithmic Properties Involving One:Basic Logarithmic Properties Involving One: loglogbbb = 1 because bb = 1 because b11 = b. = b. loglogbb1 = 0 because b1 = 0 because b00 = 1 = 1

Inverse Properties of Logarithms:Inverse Properties of Logarithms: loglogbbbbx x = x because b= x because bxx = b = bxx

bbloglogbbxx = x because b raised to the log of some = x because b raised to the log of some

number x (with the same base) equals that number x (with the same base) equals that numbernumber

Characteristics of GraphsCharacteristics of Graphs

The x-intercept is (1,0). The x-intercept is (1,0). There is no y-intercept.There is no y-intercept.

The y-axis is a vertical The y-axis is a vertical asymptote; x = 0.asymptote; x = 0.

Given logGiven logbb(x), If b > 1, the (x), If b > 1, the

function is increasing. If function is increasing. If 0<b<1, the function is 0<b<1, the function is decreasing.decreasing.

The graph is smooth and The graph is smooth and continuous. There are no continuous. There are no sharp corners or gaps.sharp corners or gaps.

TransformationsTransformationsVertical Shift Vertical Shift

Vertical shiftsVertical shifts– Moves the same as all Moves the same as all

other functions!other functions!– Added or subtracted Added or subtracted

from the whole function from the whole function at the end (or at the end (or beginning)beginning)

TransformationsTransformationsHorizontal ShiftHorizontal Shift

Horizontal shiftsHorizontal shifts– Moves the same as all Moves the same as all

other functions!other functions!– Must be “hooked on” to Must be “hooked on” to

the x value!the x value!

TransformationsTransformationsReflectionsReflections

g(x)= - logg(x)= - logbbxx Reflects about the x-axisReflects about the x-axis

g(x) = logg(x) = logbb(-x)(-x) Reflects about the y-axisReflects about the y-axis

TransformationsTransformationsVertical Stretching and ShrinkingVertical Stretching and Shrinking

f(x)=f(x)=c c loglogbbxx

Stretches the graph if Stretches the graph if the c > 1the c > 1

Shrinks the graph if Shrinks the graph if

0 < c < 10 < c < 1

TransformationsTransformationsHorizontalHorizontal Stretching and Shrinking Stretching and Shrinking

f(x)=logf(x)=logbb(cx)(cx)

Shrinks the graph if the Shrinks the graph if the c > 1c > 1

Stretches the graph if Stretches the graph if

0 < c < 10 < c < 1

DomainDomain

Because a logarithmic function reverses the Because a logarithmic function reverses the domain and range of the exponential domain and range of the exponential function, the domain of a logarithmic function, the domain of a logarithmic function is the set of all positive real function is the set of all positive real numbers unless a horizontal shift is numbers unless a horizontal shift is involved.involved.

0,

Domain Con’t.Domain Con’t.

Domain

2,

Domain

0,

Domain

4,

Properties of Commons LogsProperties of Commons Logs

General General PropertiesProperties

Common Common LogarithmsLogarithms

(base 10)(base 10)

loglogbb1 = 0 1 = 0 log 1 = 0 log 1 = 0

loglogbbb = 1 b = 1 log 10 = 1 log 10 = 1

loglogbbbbx x = x = x log 10log 10xx = x = x

bbloglogbb

xx = x = x 1010logxlogx = x = x

Properties of Natural LogarithmsProperties of Natural Logarithms

General General PropertiesProperties

Natural Natural LogarithmsLogarithms

(base e)(base e)

loglogbb1 = 0 1 = 0 ln 1 = 0 ln 1 = 0

loglogbbb = 1 b = 1 ln e = 1 ln e = 1

loglogbbbbx x = x = x ln eln exx = x = x

bbloglogbb

xx = x = x eelnxlnx = x = x