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Transcript of 1 C ollege A lgebra Inverse Functions ; Exponential and Logarithmic Functions (Chapter4) L:18 1...
1
College AlgebraInverse Functions ; Exponential
and Logarithmic Functions(Chapter4)
L:18
1
University of PalestineIT-College
/ /١٤٤٤ ٠٩ ٢٩
The Natural Base eAn irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately,
The number e is called the natural base. The function f (x) = ex is called the natural exponential function.
2.71828...e
-1
f (x) = ex
f (x) = 2x
f (x) = 3x
(0, 1)
(1, 2)
1
2
3
4
(1, e)
(1, 3)
5
Solving Exponential Equations, where x is in the exponent, BUT the bases DO NOT MATCH.
Step 1: Isolate the exponential expression.
Get your exponential expression on one side everything outside of the exponential expression on the other side of your equation.
Step 2: Take the natural log of both sides.
The inverse operation of an exponential expression is a log. Make sure that you do the same thing to both sides of your equation to keep them equal to each other.
Step 3: Use the properties of logs to pull the x out of the exponent.
Step 4: Solve for x.
Now that the variable is out of the exponent, solve for the variable using inverse operations to complete the problem.
6
Example 1: Solve the exponential equation Round your answer to two decimal places.
Step 1: Isolate the exponential expression.
This is already done for us in this problem.
Step 2: Take the natural log of both sides.
Step 3: Use the properties of logs to pull the x out of the exponent.
8
Example 2: Solve the exponential equation Round your answer to two decimal places.
Step 1: Isolate the exponential expression.
Step 2: Take the natural log of both sides.
Step 3: Use the properties of logs to pull the x out of the exponent.
10
Example 3: Solve the exponential equation Round your answer to two decimal places.
Step 1: Isolate the exponential expression.
Step 2: Take the natural log of both sides.
Step 3: Use the properties of logs to pull the x out of the exponent.
12
Exponential Equations
•Solve 1
814
.
x
1
3
181
4
(4 ) 81
4 81
3
3
Def. of negative exponent
( )
4 4 Write 81 as a power of 4
Property (b)
Multiply b 1 y
mx
x
mn
x
n
x
a
x
a
x
13
Another Example
•Solve 3x + 1 = 27x 3
1 3
1 3
1 3 9
3
27
3 Write 27 as a power of 3.
(
3
3 ( )
3 3
1 3 9
2 10
5
)
Property (b)
Subtract 3 and 1.
Divide by 2.
x x
x x
x nx m mn
x x
x
x
a a
x
Objectives:After completing this tutorial, you should be able to:
1. Know the definition of a logarithmic function. 2. Write a log function as an exponential function and vice
versa. 3. Graph a log function. 4. Evaluate a log. 5. Be familiar with and use properties of logarithms in
various situations.6. Solve logarithmic equations.
sections 4.5,4.6,4.7& Equations Logarithmic Functions
Definition of Log Function
For all real numbers y, and all positive numbers a (a > 0) and x, where a 1:
Meaning of logax
A logarithm is an exponent; logax is the exponent to which the base a must be raised to obtain x.
(Note: Logarithms can be found for positive numbers only)
A LOG IS ANOTHER WAY TO WRITE AN EXPONENT.
log if and only if .yay x x a
Location of Base and Exponent in Exponential and Logarithmic Forms
Logarithmic form: y = logb x Exponential Form: by = x. Logarithmic form: y = logb x Exponential Form: by = x.
Exponent Exponent
Base Base
Example : Express the logarithmic equation exponentially
We want to use the definition that is above: if and only if .
Examples
Write each equation in its equivalent exponential form.a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y
Solution With the fact that y = logb x means by = x,
c. log3 7 = y or y = log3 7 means 3y = 7.
a. 2 = log5 x means 52 = x.
Logarithms are exponents.
Logarithms are exponents.
b. 3 = logb 64 means b3 = 64.
Logarithms are exponents.
Logarithms are exponents.
Evaluating Logs
Step 1: Set the log equal to x.
Step 2: Use the definition of logs shown above to write the equation in exponential form.
Step 3: Find x.
Whenever you are finding a log, keep in mind that logs are another way to write exponents. You can always use the definition to help you evaluate.
Evaluatea. log2 16 b. log3 9 c. log25 5
Solution
log25 5 = 1/2 because 251/2 = 5.25 to what power is 5?c. log25 5
log3 9 = 2 because 32 = 9.3 to what power is 9?b. log3 9
log2 16 = 4 because 24 = 16.2 to what power is 16?a. log2 16
Logarithmic Expression Evaluated
Question Needed for Evaluation
Logarithmic Expression
Text Example
Characteristics of the Graph of f(x) = logax
The points (1, 0), and (a, 1) are on the graph.
If a > 1, then f is an increasing function; if 0 < a < 1, then f is a decreasing function.
The y-axis is a vertical asymptote.
The domain is (0, ), and the range is (, ).
1, 1 ,
a
Example
Graph Write
in exponential form
as
Now find some ordered pairs.
1/ 4( ) logf x x
1/ 4( ) logf x y x
1
4
y
x
14
21/16
01
yx
Graph Write
in exponential form
as
Now find some ordered pairs.
5( ) logf x x
10.2
15
01
yx
5yx
5( ) logf x x
Example
Translated Logarithmic Functions
Graph the function.
The vertical asymptote is x = 1.
To find some ordered pairs, use the equivalent exponent form.
3( ) log ( 1) f x x
3log ( 1)
1 3
3 1
y
y
y x
x
x
Translated Logarithmic Functions continued
Graph To find some ordered
pairs, use the equivalent exponent form.
4( ) (log ) 1 f x x
4
4
1
log 1
1 log
4 y
y x
y x
x
Properties of Logarithms, For x > 0, y > 0, a > 0, a 1, and any real number r:
The logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number.
Power Property
The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers.
Quotient Property
The logarithm of a product of two numbers is equal to the sum of the logarithms of the numbers
Product Property
DescriptionProperty
log log loga a axy x y
log log loga a a
xx y
y
log logra ax r x
Using the Properties of Logarithms
Rewrite each expression. Assume all variables represent positive real numbers with a 1 and b 1.
a)
b)
c)
6
12log
7
4log 11
2loga
abc
w
6 6 6log log12
12 77
log
4 4 41/ 2log 11 log (11) lo
1
2g 11
22
log log log log log
log log log 2log
a a a a a
a a a a
abca b c w
wa b c w
Using the Properties of Logarithms
Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with a 1 and b 1.
a)
b)
3 3 3log ( 1) log log 5x x
3log 5logy yr t
3 3 3 3log log log log( 1)
1 55
( )x x
x x
3
3 5
5
log log log log
l
53
og
y y y y
y
r r
r
t
t
t
Using the Properties of Logarithms
Expand as much as possible. Evaluate without a calculator where possible
Inverse Properties of Logarithms
For a > 0, a 1:
By the results of this theorem:
log and log .a x xaa x a x
7 1log 07 10 35log 5 3
Inverse Property I
Inverse Properties of Logarithms
For b > 0, b 1:
By the results of this theorem:
Inverse Property II
,
b logb x = x
Basic Logarithmic Properties Involving One
Logb b = 1
because 1 is the exponent to which b must be raised to obtain b. (b1 = b).
Logb 1 = 0
because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).
Properties of Common Logarithms
General Properties Common Logarithms
1. logb 1 = 0 1. log 1 = 0
2. logb b = 1 2. log 10 = 1
3. logb bx = x 3. log 10x = x4. b logb x = x 4. 10 log x = x
log b b = 1 log b 1 = 0 log 4 4 = 1
log 8 1 = 0 3 log 3 6 = 6 log 5 5 3 = 3
2 log 2 7 = 7
Examples of Logarithmic Properties
Natural Logarithms
Logarithms with a base of e are referred to a natural logarithms.
So if f(x) = ex , then f(x) = loge x = lnx Recall, e = 2.71828
Properties of Natural Logarithms
General NaturalProperties Logarithms
1. logb 1 = 0 1. ln 1 = 0
2. logb b = 1 2. ln e = 1
3. logb bx = 0 3. ln ex = x4. b logb x = x 4. e ln x = x
Examples
log e e = 1
log e 1 = 0
e log e 6 = 6
log e e 3 = 3
Change-of-Base Theorem
For any positive real numbers x, a, and b, where a 1 and b 1:
logax =lnx/ lna
loglog .
logb
ab
xx
a
Examples
a) log512 b) log2.4
Use the change-of-base theorem to find an approximation to four decimal places for each logarithm.
5
lnlog
ln2.4849
1.60941.5440
25
121
2
loglog
log
.3979
.3010
2
1.321
..4
9
4
Solving Logarithmic Equations
Solve each equation. a) b)
3
33
27log 3
64
27
64
3
4
3
4
x
x
x
x
9
3 / 2
1/ 2 3
3
3log
2
9
(9 )
3
27
x
x
x
x
x
Example
Solve 8x = 15
The solution set is {1.3023}.
8 15
8 15
ln8 ln15
ln15
ln
ln
ln
81.3023
x
x
x
x
x
Example
Solve continued 2 1 35 .3 x x
2 1 3
2 1 3
5 .3
5 .3
(2 1)ln5 ( 3)ln.3
2 ln5 ln5 ln.3 3ln.3
2 ln5 ln.3 3ln.3 ln5
(2ln5 ln.3) 3l
ln ln
n.3 ln5
x x
x x
x x
x x
x x
x
3ln.3 ln5
2ln5 ln.3ln.027 ln5
ln 25 ln.3ln.135
25ln
.3.4528
x
x
x
x
Solve log ( 2) log ( 1) log ( 2) b b bx x x
2
2
log ( 2) log ( 1)( 2)
( 2) 3 2
0 4
0 ( 4)
0 4 0
0 4
b bx x x
x x x
x x
x x
x or x
x x
The only valid solution is x = 4.
Example
Example
Solve 2
2
2
5log log
l
84
58
48( 4) 5
8 32 5
9
og (
2
5 ) log 4 3
3
(
7
)
x
x
x
x
xx x
x x
x
x
x x
Example
Solve continued
2
2
4ln ln(8 )
1
48
1(8 )( 1) 4
8
ln( 4) ln( 1) ln )
4
9
(
8 4
8
8
xx
x
xx
xx x x
x x x
x x
x
x
x
x x 20 8 12
0 ( 6)( 2)
6 0 2 0
6 2
x x
x x
x or x
x or x
The only valid solution is x = 2.
Properties of Logarithms
loga a 1 a MaMlog loga
ra r
log log loga a aMN M N
log log loga a aMN
M N
log loga aNN
1
log logar
aM r M
1
3log
2
2
x
xxa
Write the following expression as the sum and/or difference of logarithms. Express all powers as factors.
log log loga a a
x x
xx x x
2
22 23
13 1
log log loga a ax x x2 23 1
212
3 2 1log log loga a ax x x
Write the following expression as a single logarithm.
1log12log4
1log3 xxx aaa
log log loga a ax x x3 142 1 1
log log loga a ax x x3 4 2 1 1
log loga ax x x3 4 2 1 1
loga
x xx
3 4 2 11
Most calculators only evaluate logarithmic functions with base 10 or base e. To evaluate logs with other bases, we use the change of base formula.
loglogloga
b
b
MMa
loglog
Ma
lnln
Ma
Calculate log5 63
logloglog5 63
635
lnln
635
2 574.
Solve: log log6 63 2 1x x log log6 63 2 1x x
log6 3 2 1x x
log62 6 1x x
x x2 16 6 x x2 16 6 x x2 12 0
x x 4 3 0x x 4 3 or
Solution set: {x | x = 3}
Solve: 9 3 10 02x x
9 3 10 02x x
3 3 3 10 02 2x x
3 9 3 10 02x x
3 10 3 1 0x x
3 10 0 3 1 0x x or
3 10 3 1x x or No Solution x 0
Solution set: x x 0
Solve: 7 53 2x x
7 53 2x x
ln ln7 53 2x x
x xln ln7 3 2 5 x xln ln ln7 3 5 2 5
x xln ln ln7 3 5 2 5 x ln ln ln7 3 5 2 5
x
2 5
7 3 5
ln
ln ln1117.