Logarithmic Functions(1) (1)
-
Upload
thomas-wash -
Category
Documents
-
view
217 -
download
0
Transcript of Logarithmic Functions(1) (1)
-
7/31/2019 Logarithmic Functions(1) (1)
1/20
1
Logarithmic Functions
-
7/31/2019 Logarithmic Functions(1) (1)
2/20
2
Logarithmic Functions
In this section, another type of function will be studied called
the logarithmic function. There is a close connection
between a logarithmic function and an exponential function.
We will see that the logarithmic function and exponentialfunctions are inverse functions.
-
7/31/2019 Logarithmic Functions(1) (1)
3/20
3
Logarithmic Functions
The logarithmic function with base two is defined to be the
inverse of the one to one exponential function
Notice that the exponential
function
is one to one and therefore has
an inverse.
0
1
2
3
4
5
6
7
8
9
-4 -2 0 2 4
graph of y = 2^(x)
approaches the negative x-axis as x gets
large
passes through (0,1)
2
x
y
2x
y
-
7/31/2019 Logarithmic Functions(1) (1)
4/20
4
Inverse of an Exponential Function
Start with
Now, interchangex andy coordinates:
There are no algebraic techniques that can be used to solve for
y, so we simply call this functiony the logarithmic function
with base 2. The definition of this new function is:
if and only if
2xy
2yx
2log x y 2yx
-
7/31/2019 Logarithmic Functions(1) (1)
5/20
-
7/31/2019 Logarithmic Functions(1) (1)
6/20
-
7/31/2019 Logarithmic Functions(1) (1)
7/20
7
Logarithmic-Exponential
Conversions
Study the examples below. You should be able to convert a
logarithmic into an exponential expression and vice versa.
1.
2.
3.
4.
4log (16) 4 16 2xx x
3125 5 5log 125 3
1
281
181 9 81 9 log 9
2
3
3 3 33
1 1log ( ) log ( ) log (3 ) 3
27 3
-
7/31/2019 Logarithmic Functions(1) (1)
8/20
8
Solving Equations
Using the definition of a logarithm, you can solve equations
involving logarithms. Examples:
3 3 3log (1000) 3 1000 10 10b b b b
56log 5 6 7776x x x
In each of the above, we converted from log form to
exponential form and solved the resulting equation.
-
7/31/2019 Logarithmic Functions(1) (1)
9/20
9
Properties of Logarithms
These are the properties of logarithms.MandNare positive real
numbers, b not equal to 1, andp andx are real numbers.(For 4, we needx > 0).
5. log log log
6. log log log
7. log log
8. log log
b b b
b b b
p
b b
b b
MN M N
MM N
N
M p M
M N iff M N
log
1.log (1) 02.log ( ) 1
3.log
4. b
b
b
x
b
x
b
b x
b x
-
7/31/2019 Logarithmic Functions(1) (1)
10/20
10
Solving Logarithmic Equations
1. Solve forx:
-
7/31/2019 Logarithmic Functions(1) (1)
11/20
11
Solving Logarithmic Equations
1. Solve forx:
2. Product rule
3. Special product
4. Definition of log
5. x can be +10 only
6. Why?
4 4
4
2
4
3 2
2
2
log ( 6) log ( 6) 3
log ( 6)( 6) 3
log 36 34 36
64 36
100
10
10
x x
x x
xx
x
x
x
x
-
7/31/2019 Logarithmic Functions(1) (1)
12/20
12
Another Example
1. Solve:
-
7/31/2019 Logarithmic Functions(1) (1)
13/20
13
Another Example
1. Solve:
2. Quotient rule
3. Simplify
(divide out common factor )
4. rewrite
5 definition of logarithm
6. Property of exponentials
-
7/31/2019 Logarithmic Functions(1) (1)
14/20
14
Common Logs and Natural Logs
Common log Natural log
10log logx x ln( ) logex x
2.7181828e If no base is indicated,the logarithm is
assumed to be base 10.
-
7/31/2019 Logarithmic Functions(1) (1)
15/20
15
Solving a Logarithmic Equation
Solve forx. Obtain the exact
solution of this equation in terms
of e (2.71828)
ln (x + 1)lnx = 1
-
7/31/2019 Logarithmic Functions(1) (1)
16/20
16
Solving a Logarithmic Equation
Solve forx. Obtain the exact
solution of this equation in terms
of e (2.71828)
Quotient property of logs
Definition of (natural log)
Multiply both sides byx
Collectx terms on left side
Factor out common factorSolve forx
ln (x + 1)lnx = 1
xe =x + 1
xe - x = 1x (e-1) = 1
1
1x
e
x
xe
11
11
ln
x
x
-
7/31/2019 Logarithmic Functions(1) (1)
17/20
17
Application
How long will it take money to double
if compounded monthly at 4 %
interest?
-
7/31/2019 Logarithmic Functions(1) (1)
18/20
18
Application
How long will it take money to double
if compounded monthly at 4 %
interest?
1. Compound interest formula
2. ReplaceA by 2P (double the
amount)
3. Substitute values for r and m
4. Divide both sides by P
5. Take ln of both sides6. Property of logarithms
7. Solve for tand evaluate expression
Solution:
12
12
12
1
0.042 1
12
2 (1.003333...)
ln 2 ln (1.003333...)
ln 2 12 ln(1.00333...)
ln 217.36
12ln(1.00333...)
mt
t
t
t
rA P
m
P P
t
t t
-
7/31/2019 Logarithmic Functions(1) (1)
19/20
19
Logarithmic functions compared to
others
Among increasing functions, the logarithmic functions
with bases b > 1 increase much more slowly for largevalues ofx than either exponential or polynomial
functions. When a visual inspection of the plot of a
data set indicates a slowly increasing function, a
logarithmic function ofter provides a good model.
-
7/31/2019 Logarithmic Functions(1) (1)
20/20
Comparing functions
20
beat
beat
Logarithmic functions