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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Exponential and Logarithmic Functions 5 Exponential Functions Logarithmic Functions.

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Presentation on theme: "Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Exponential and Logarithmic Functions 5 Exponential Functions Logarithmic Functions."— Presentation transcript:

1 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Exponential and Logarithmic Functions 5 Exponential Functions Logarithmic Functions Compound Interest Models

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. Domain: All reals Range: y > 0 (0,1) x y

3 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Laws of Exponents LawExample

4 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Properties of Exponential Functions 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.

5 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Examples Ex.Simplify the expression. Ex.Solve the equation

6 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Logarithms An logarithmic of x to the base b is defined by Ex.

7 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Examples Ex. Solve each equation a. b.

8 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Notation: Common Logarithm Natural Logarithm Laws of Logarithms

9 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Example Use the laws of logarithms to simplify the expression:

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, ) 2.Range: 3. Intercept: (1, 0) 4. Continuous on (0, ) 5. Increasing on (0, ) if b > 1 Decreasing on (0, ) if b < 1

11 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Logarithmic Function Graphs Ex. (1,0)

12 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Solve Apply ln to both sides.

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

14 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. = $

15 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Effective Rate of Interest r eff = 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.

16 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. 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

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.

18 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Continuous Compound Interest A = the accumulated amount after t years P = Principal r = Nominal interest rate per year t = Number of years

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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