LOGARITHMS Section 4.2
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Transcript of LOGARITHMS Section 4.2
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