Exponential and Logarithmic Functions
5
• Exponential Functions
• Logarithmic Functions
• Compound Interest
• Models
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Exponential Function
An exponential function with base b and exponent x
is defined by
f ( x)  b
Ex. f ( x)  3x
x
y
1
1
0
1
2
1
3
9
x
b  0, b  1
y  f ( x)
Domain: All reals
Range: y > 0
3
(0,1)
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Laws of Exponents
Law
Example
1. b x  b y  b x  y
bx
x y
2. y  b
b
 
3. b
x
y
b
xy
4.  ab   a b
x
x x
x
x
a
a
5.    x
b
b
2 2  2  2  8
12
5
123
9
5
5
3
5
6
1
1/ 3
6 / 3
2
8
8
8 
64
1/ 2
5/ 2
6/ 2
3
 
 2m 
3
 2 m  8m
1/ 3
 8 
 
 27 
3
3
3
81/ 3
2
 1/ 3 
3
27
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Properties of Exponential
x
f ( x) 
b
Functions
b  0, b  1
1. The domain is  ,  .
2. The range is (0,  ).
3. It passes through (0, 1).
4. It is continuous everywhere.
5. If b > 1 it is increasing on  ,   .
If b < 1 it is decreasing on  ,   .
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Examples
Ex. Simplify the expression.
3x y 
2 1/ 2
x3 y 7
4
34 x8 y 2 81x5
 3 7  5
y
x y
Ex. Solve the equation
43 x 1  24 x 2
23 x1
4 x 2
2
2
26 x  2  2 4 x  2
6x  2  4x  2
2 x  4
x  2
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Logarithms
An logarithmic of x to the base b is defined by
y  logb x if and only if x  b
y
 x  0
Ex. log 3 81  4
log 7 1  0
log1/ 3 9  2
log 5 5  1
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Examples
Ex. Solve each equation
a. log 2 x  5
x  25  32
b. log 27 3  x
3  27
3  33 x
1  3x
1
x
3
x

am  an  m  n

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Laws of Logarithms
1. l og b mn  log b m  log b n
m
2. log b    log b m  log b n
n
3. logb m n  n logb m
4. log b 1  0
5. logb b  1
Notation:
Common Logarithm log x  log10 x
Natural Logarithm ln x  log e x
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Example
Use the laws of logarithms to simplify the
expression:
7
25 x y
log5
z
 log5 25  log5 x7  log5 y  log5 z1/ 2
1
 2  7 log5 x  log5 y  log 5 z
2
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Logarithmic Function
An logarithmic function of x to the base b is defined
by
f ( x)  logb x
b  0, b  1
Properties
1.
2.
3.
4.
5.
Domain: (0,  )
Range:  ,  
Intercept: (1, 0)
Continuous on (0,)
Increasing on (0, ) if b > 1
Decreasing on (0,  ) if b < 1
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Logarithmic Function Graphs
Ex.
f ( x)  log3 x
y3
f ( x)  log1/ 3 x
1
y 
 3
x
x
(1,0)
y  log3 x
y  log1/ 3 x
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
x
e and ln x
e
x
ln x
ln e  x
x
 x  0
 for any real number x 
1 2 x 1
Ex. Solve e
 10
3
e2 x1  30
2 x  1  ln(30)
Apply ln to both sides.
ln(30)  1
x
 1.2
2
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Compound Interest
r

A  P 1  
 m
mt
A = the accumulated amount after mt periods
P = Principal
r = Nominal interest rate per year
m = Number of periods/year
t = Number of years
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Example
Find the accumulated amount of money after 5
years if $4300 is invested at 6% per year
compounded quarterly.
r

A  P 1  
 m
mt
 .06 
A  4300 1 

4 

4(5)
= $5791.48
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Effective Rate of Interest
m
r

reff  1    1
 m
reff = Effective rate of interest
r = Nominal interest rate/year
m = number of conversion periods/year
Ex. Find the effective rate of interest corresponding to
a nominal rate of 6.5% per year, compounded monthly.
m
reff
12
r

 .065 
 1    1  1 
  1  .06697
12 

 m
So about 6.67% per year.
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Present Value for Compound
Interest
r

P  A 1  
 m
 mt
A = the accumulated amount after mt periods
P = Principal
r = Nominal interest rate per year
m = Number of periods/year
t = Number of years
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Example
Find the present value of $4800 due in 6 years
at an interest rate of 9% per year compounded
monthly.
 mt
r

P  A 1  
 m
 .09 
P  4800 1 

12


12(6)
 $2802.83
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Continuous Compound Interest
A  Pe
rt
A = the accumulated amount after t years
P = Principal
r = Nominal interest rate per year
t = Number of years
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Example
Find the accumulated amount of money after
25 years if $7500 is invested at 12% per year
compounded continuously.
A  Pe
rt
A  7500e
0.12(25)
 $150, 641.53
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .