1. Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Exponential and Logarithmic Functions 5
Exponential Functions
Logarithmic Functions
Compound Interest
Models
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2. 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
Ex.
x y
Domain: All reals
Range: y > 0
(0,1)
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Laws of Exponents
Law
Example
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4. Properties of Exponential Functions
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
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5. Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Examples
Ex.
Simplify the expression.
Ex.
Solve the equation
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Logarithms
An logarithmic of x to the base b is defined by
Ex.
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Examples
Ex. Solve each equation a. b.
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8. Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Laws of Logarithms
Notation:
Common Logarithm
Natural Logarithm
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9. Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Example
Use the laws of logarithms to simplify the expression:
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10. 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
Properties
1. Domain: (0, )
Range:
3. Intercept: (1, 0)
4. Continuous on (0, )
5. Increasing on (0, ) if b > 1
Decreasing on (0, ) if b < 1
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11. Logarithmic Function Graphs
Ex.
(1,0)
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Ex. Solve
Apply ln to both sides.
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13. Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Compound Interest
A = the accumulated amount after mt periods
P = Principal
r = Nominal interest rate per year
m = Number of periods/year
t = Number of years
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Example
Find the accumulated amount of money after 5 years if $4300 is invested at 6% per year compounded quarterly.
= $
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15. Effective Rate of Interest
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.
So about 6.67% per year.
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16. Present Value for Compound Interest
A = the accumulated amount after mt periods
P = Principal
r = Nominal interest rate per year
m = Number of periods/year
t = Number of years
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17. 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.
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18. Continuous Compound Interest
A = the accumulated amount after t years
P = Principal
r = Nominal interest rate per year
t = Number of years
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19. 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.
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