introduction to logarithms. introduction to logarithmic functions you were introduced to inverse...
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Introduction To Logarithms
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Introduction to Logarithmic Functions
You were introduced to inverse functions.•Inverse functions is the set of ordered
pair obtained by interchanging the x and y values.
f(x)
f-1(x)
GRAPHS OF EXPONENTIALS AND ITS INVERSE
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Introduction to Logarithmic Functions
•Inverse functions can be created graphically by a reflection on the y = x
axis.
y = x
f(x)
f-1(x)
GRAPHS OF EXPONENTIALS AND ITS INVERSE
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Introduction to Logarithmic Functions
•A logarithmic function is the inverse of an exponential
function
•Exponential functions have the following characteristics:
Domain: {x є R}
Range: {y > 0}
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Introduction to Logarithmic Functions
•Let us graph the exponential function y = 2x
•Table of values:
GRAPHS OF EXPONENTIALS AND ITS INVERSE
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Introduction to Logarithmic Functions
•Let us find the inverse the exponential function
y = 2x•Table of values:
GRAPHS OF EXPONENTIALS AND ITS INVERSE
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Introduction to Logarithmic Functions
•When we add the function f(x) = 2x to this graph, it is evident that the inverse is a reflection on the y = x axis
f-1(x)
f(x)
GRAPHS OF EXPONENTIALS AND ITS INVERSE
f(x) f-
1(x)
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Introduction to Logarithmic Functions
•You will find the inverse of an exponential
algebraically•Write the process in your notes
FINDING THE INVERSE OF AN EXPONENTIAL
y = ax
Interchange x yx = ay
•We write these functions as:x = ay
y = logax
x = ay
base
exponent
y = logaaxx
baseexponent
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Introduction to Logarithmic Functions
FINDING THE INVERSE OF AN EXPONENTIAL
y=axExponential
FunctionInverse of the Exponential
Function
x ylogLogarithmic Form
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Introduction to Logarithmic Functions
CHANGING FORMSExample 1) Write the following into logarithmic form:a) 43 =
64
b) 45 = 256
c) 27 = 128
d) (1/3)x=27
4log 64 3
4log 256 5
2log 128 7
13
log 27 x
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Introduction to Logarithmic Functions
CHANGING FORMSExample 2) Write the following into exponential form:
a) log264=6
b) log255=1/2
c) log81=0
d) log1/39=2
ANSWERS
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Introduction to Logarithmic Functions
CHANGING FORMSExample 2) Write the following into exponential form:
a) log264=6
26 = 64
251/2 = 5
80 = 1
(1/3)2 = 1/9
b) log255=1/2c) log81=0
d) log1/39=2
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Introduction to Logarithmic Functions
EVALUATING LOGARITHMSExample 3) Find the value of x for each example:
a) log1/327 = xb) log5x = 3
c) logx(1/9) = 2d) log3x = 0
ANSWERS
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Introduction to Logarithmic Functions
EVALUATING LOGARITHMSExample 3) Find the value of x for each example:
a) log1/327 = x b) log5x = 3
c) logx(1/9) = 2
d) log3x = 0
(1/3)x = 27(1/3)x = (1/3)-3
x = -3
53 = xx = 125
x2 = (1/9)
x = 1/3
30 = xx = 1
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Introduction to Logarithmic Functions
BASE 10 LOGSScientific calculators can perform logarithmic operations. Your calculator has
a LOG button.This button represents logarithms in BASE 10 or
log10Example 4) Use your calculator to find the value of each of the following:
a) log101000 b) log 50
c) log -1000=
3
= 1.69
9
= Out of Domain
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What is a logarithm ?What is a logarithm ?
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Solution: log2 8 3
We read this as: ”the log base 2 of 8 is equal
to 3”.
3Write 2 8 in logarithmic form.
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Write 42 16 in logarithmic form.
Solution: log4 16 2
Read as: “the log base 4 of 16 is equal to 2”.
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Solution:
Write 2 3 1
8 in logarithmic form.
log2
1
8 3
1Read as: "the log base 2 of is equal to -3".
8
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Okay, so now it’s time for you to try some on
your own.
1. Write 72 49 in logarithmic form.
7Solution: log 49 2
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log5 10Solution:
2. Write 50 1 in logarithmic form.
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3. Write 10 2 1
100 in logarithmic form.
Solution: log10
1
100 2
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Solution: log16 4 1
2
4. Finally, write 161
2 4
in logarithmic form.
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Write log3 814 in exponential form
Solution: 34 81
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Write log2
1
8 3 in exponential form.
Solution: 2 3 1
8
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Okay, now you try these next three.
1. Write log10 100 2 in exponential form.
3. Write log27 3 1
3 in exponential form.
2. Write log5
1
125 3 in exponential form.
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1. Write log10 100 2 in exponential form.
Solution: 102 100
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2. Write log5
1
125 3 in exponential form.
Solution: 3 15
125
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3. Write log27 3 1
3 in exponential form.
Solution: 271
3 3
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Our final concern then is to determine why logarithms
like the one below are undefined.
Our final concern then is to determine why logarithms
like the one below are undefined.
Can anyone give us an
explanation ?
Can anyone give us an
explanation ?
2log ( 8)
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One easy explanation is to simply rewrite this logarithm in exponential
form. We’ll then see why a negative value
is not permitted.First, we write the problem with a variable.
2y 8 Now take it out of the logarithmic form
and write it in exponential form.What power of 2 would gives us -8 ?23 8 and 2 3
1
8
Hence expressions of this type are undefined.
2log ( 8) undefined WHY?
2log ( 8) y