name : ______________ ( ) class : ________ date :_________
DESCRIPTION
Name : ______________ ( ) Class : ________ Date :_________. Unit 7: Logarithmic and Exponential Functions. Objectives:. Graphs. Logarithms. Common and Natural Logarithms. Laws of Logarithms. Logarithmic Equations. Solving Equations of the Form. Graphs of Exponential Functions. - PowerPoint PPT PresentationTRANSCRIPT
Name : ______________ ( ) Class : ________ Date :_________
Objectives:
Unit 7: Logarithmic and Exponential Functions
Graphs
Solving Equations of the Form
Logarithmic Equations
Laws of Logarithms
Logarithms
Common and Natural
Logarithms
xa b
Graphs of Exponential Functions
Graph ofxy a
x
y
-3
-2
-1
1
2
3
4
5
6
7
Graph ofxy e
Graphs of Exponential Functions
x
y
-6
-5
-4
-3
-2
-1
1
2
3
4
Graph of
lny x
x
y
-6
-5
-4
-3
-2
-1
1
2
3
4
Graph of
y = lg x
Graphs of Common Logarithms
Convert to logarithmic form.
3
1log 2
9
2 13
9 The logarithm or
index for the given base is -2.
Convert to index form.
4log 64 3
34 64
The base is 3.
The base is 4. The logarithm or index for the given
base is 3.
Logarithms
If a logarithm is defined for base a, then
log 1a a
and
log 1 0a
1since a a
0since 1a
Special Cases
Logarithms
Evaluate the following.
2 5log 1 4log 5
2
5
3log 2
4 2log 1x x
0 4 1
4
23
2
1
0
2
4
25 25
4 16
Logarithms
Example 1:
A common logarithm is a logarithm to the base 10.
10log is commonly abbreviated to lgx x
Tables of common logarithms were often used for calculating in
the days before the electronic calculator.
On a scientific calculator, common logarithms can be evaluated using the LOG key.
Logarithms
A natural logarithm is a logarithm to the base e.
elog is commonly abbreviated to lnx x
Natural logarithms are also known as
Naperian logarithms after
John Napier (1550 - 1617).
On a scientific calculator, natural logarithms can be evaluated using the LN key.
Logarithms
From the definitions of logarithms, the following statements are equivalent.
lg 10xy x y
Let’s use these definitions in some
examples.
and
ln exy x y
Logarithms
Convert the following to index form.
lg1000 3
ln 3 m
3 1000 10
3 em
Convert the following to logarithmic form.
2 10 0.01 lg 0.01 2
2 e x k ln 2k x
Index form
Index form
Logarithmic form
Logarithmic form
Logarithms
Example 2:
Find y in terms of x.
ln 1y x
1 exy
2lg 2y x
2lg
2
xy
21
2 210 10x x
y
e 1xy
Index form
Index form
Rearrange
Rearrange
Alternate form
Surds, Indices and Logarithms
(a)
(b)
Example 3:
Solve for x.
ln 4 lg3 lg5x
ln 4 0.4771 0.6990x
0.33354 ex 4 1.396x
0.349x
ln 4 0.3335x
Evaluate using the calculator.
Index form
In most calculators, the function ex is on
the same key as LN.
Evaluate and solve for x.
Logarithms
Example 4:
Solve for x.
2lg 2 lg3x
2 lg 2 0.4771x
0.22762 10x 2 1.689x
3.69x
lg 2 0.2276x
Evaluate using the calculator.
Index form
In most calculators, the function 10x is
on the same key as LOG.
Evaluate and solve for x.
Logarithms
Example 5:
The Power Law
If and are positive numbers and 1, then
log log for any real number .ra a
a x a
x r x r
If , and are positive numbers and 1, then
log log log .a a a
a x y a
xy x y
The Product Law
If , and are positive numbers and 1, then
log log log .a a a
a x y a
xx y
y
The Quotient Law
Logarithms
Let’s use these laws in some
examples.
If , and are positive numbers and 1, 1, then
loglog .
logc
ac
a b c a c
bb
a
The Change of Base Law
1log .
logab
ba
A special case
Logarithms
Example 6:
Evaluate the following.
3 3 3log 4 log 2 log 72
Combine using the product and quotient laws.
2log 5 3log 2 log 4x x x
3
4 2log
72
23log 3
32log 32
2 3log 5 log 2 log 4x x x 2
3
5 4log
2x
25log
2x
Combine using the product and quotient laws.
Apply the power law.Apply the
power law.
Logarithms
(a)
(b)
4 4Given that log 3 and log 5
express the following in terms of and .
a b
a b
Separate using the product and quotient laws.
4log 45 24og 5l 3
24 4log 3 log 5
Apply the power law.
4 42log 3 log 5
2a b
Logarithms
Example 7:
2 23 log log 2 x y x y
Combine, applying the quotient law.
2 23 log 2 logx y x y
2
23 log
x y
x y
Rearrange and solve the equation.
322
x y
x y
Find y in terms of x.
28
x y
x y
2 8 8x y x y
10 7y x 0.7y x
Arrange the log terms on one
side.
Index form
Logarithms
Example 8:
Apply the power law.
lg 4 lg10lg5 lg 2
lg 10lg 25
Apply the
change of base law.
Evaluate the following.
5 2
25
log 4 log 10
log 10
lg 4 lg10 lg 25
lg5 lg 2 lg 10
12
2 2lg 2 lg10 lg5
lg5 lg 2 lg10
12
2lg 2 lg10 2lg5
lg5 lg 2 lg10 8
Express as powers of 2, 5
and 10.
Logarithms
Example 9:
log loga aM N M N
For two logarithms of the same base,
Let’s use this property to solve some logarithmic
equations.
Logarithms
An Important Property of Logarithms
Combine using the
product law.
3 3 3log 2 log 2 log 2 1x x x
Example : 10
3 3log 2 2 log 2 1x x x
23 3log 4 log 2 1x x
2 4 2 1x x 2 2 3 0x x
1, 3x x 3 3But, if 1, then log 2 and log 2 1
are undefined as 2 0 and 2 1 0
x x x
x x
Use the property of logarithms.
Remember to check if the results
are acceptable
.
3 1 0x x
Logarithms
So, x = 3.
Apply the power law.
Example 11:
Solve the following equation.
Index form
Remember to check if the results are acceptable.
4 2 9log 6 log 8 log 3x
123
4 2 9log 6 log 2 log 9x
4log 6 3 0.5x
4log 6 3.5x 3.56 4x
122x 6 128x
Logarithms
log4(6 – x) is defined for x = –122.
Apply the change of base law.
Example 12 :
Solve the following equation.
Substitute
Both results are
acceptable.
3log 2 3log 3xx
3
13 loglog 2 3 xx
3Let log x u 3then 2 uu 2 2 3u u
2 2 3 0u u 3 1 0u u
3 and 1u u 3 13 and 3x x
127 and 3x x
Logarithms
log loga aM N M N
For two logarithms of the same base,
Let’s use this property to solve some logarithmic
equations.
Logarithms
An Important Property of Logarithms
Combine using the
product law.
3 3 3log 2 log 2 log 2 1x x x
Example 13 :
3 3log 2 2 log 2 1x x x
23 3log 4 log 2 1x x
2 4 2 1x x 2 2 3 0x x
1, 3x x 3 3But, if 1, then log 2 and log 2 1
are undefined as 2 0 and 2 1 0
x x x
x x
Use the property of logarithms.
Remember to check if the results
are acceptable
.
3 1 0x x
Logarithms
So, x = 3.
Apply the power law.
Example 14 :
Solve the following equation.
Index form
Remember to check if the results are acceptable.
4 2 9log 6 log 8 log 3x
123
4 2 9log 6 log 2 log 9x
4log 6 3 0.5x
4log 6 3.5x 3.56 4x
122x 6 128x
Logarithms
log4(6 – x) is defined for x = –122.
Apply the change of base law.
Example 15 :
Solve the following equation.
Substitute
Both results are
acceptable.
3log 2 3log 3xx
3
13 loglog 2 3 xx
3Let log x u 3then 2 uu 2 2 3u u
2 2 3 0u u 3 1 0u u
3 and 1u u 3 13 and 3x x
127 and 3x x
Logarithms