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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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**Logarithmic Function Graphs**

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

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

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

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

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

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

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

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

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**Continuous Compound Interest**

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.

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

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GRAPHING EXPONENTIAL FUNCTIONS f(x) = 2 x 2 > 1 exponential growth 2 24–2 4 6 –4 y x Notice the asymptote: y = 0 Domain: All real, Range: y > 0.

GRAPHING EXPONENTIAL FUNCTIONS f(x) = 2 x 2 > 1 exponential growth 2 24–2 4 6 –4 y x Notice the asymptote: y = 0 Domain: All real, Range: y > 0.

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