Exponential and Logarithmic Functions 5 Exponential Functions Logarithmic

Exponential Function An exponential function with base b and exponent x is defined by

Laws of Exponents Law Example Copyright © 2006 Brooks/Cole, a division of Thomson Learning,

Properties of the Exponential Function 1. The domain is 2. The range is (0,

Examples Ex. Simplify the expression Ex. Solve the equation Copyright © 2006 Brooks/Cole, a

Logarithms The logarithm of x to the base b is defined by Ex. Copyright

Examples Ex. Solve each equation a. b. Copyright © 2006 Brooks/Cole, a division of

Laws of Logarithms Notation: Common Logarithm Natural Logarithm Copyright © 2006 Brooks/Cole, a division

Example Use the laws of logarithms to simplify the expression: Copyright © 2006 Brooks/Cole,

Logarithmic Function The logarithmic function of x to the base b is defined by

Graphs of Logarithmic Functions Ex. y y (0, 1) (1, 0) x Copyright ©

Ex. Solve Apply ln to both sides. Copyright © 2006 Brooks/Cole, a division of

Example A normal child’s systolic blood pressure may be approximated by the function where

Compound Interest Formula A = The accumulated amount after mt periods P = Principal

Example Find the accumulated amount of money after 5 years if $4300 is invested

Example How long will it take an investment of $10, 000 to grow to

Effective Rate of Interest reff = Effective rate of interest r = Nominal interest

Present Value Formula for Compound Interest A = The accumulated amount after mt periods

Example Find the present value of $4800 due in 6 years at an interest

Continuous Compound Interest Formula A = The accumulated amount after t years P =

Example Find the accumulated amount of money after 25 years if $7500 is invested

Differentiation of Exponential Functions Derivative of Exponential Function Chain Rule for Exponential Functions If

Examples Find the derivative of Find the relative extrema of + Relative Min. f

Differentiation of Logarithmic Functions Derivative of Exponential Function Chain Rule for Exponential Functions If

Examples Find the derivative of Find an equation of the tangent line to the

Logarithmic Differentiation 1. Take the Natural Logarithm on both sides of the equation and

Examples Use logarithmic differentiation to find the derivative of Apply ln Properties of ln

Exponential Growth/Decay Models A quantity Q whose rate of growth/decay at any time t

Example A certain bacteria culture experiences exponential growth. If the bacteria numbered 20 originally

Learning Curves An exponential function may be applied to certain types of learning processes

Example Suppose that the temperature T, in degrees Fahrenheit, of an object after t

Logistic Growth Model An exponential function may be applied to a logistic growth model:

Example The number of people R, in a small school district who have heard

So Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Example The number of soldiers at Fort Mac. Arthur who contracted influenza after t

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Exponential and Logarithmic Functions 5 • Exponential Functions • Logarithmic Functions • Compound Interest • Differentiation of Exponential Functions • Differentiation of Logarithmic Functions • Exponential Functions as Mathematical Models Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Exponential Function An exponential function with base b and exponent x is defined by y Ex. x 0 1 2 y 1 3 9 Domain: All reals (0, 1) Range: y > 0 x Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Properties of the Exponential Function 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 © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Logarithmic Function The logarithmic function of x to the base b is defined by Properties: 1. 2. 3. 4. 5. Domain: (0, ) Range: x-intercept: (1, 0) Continuous on (0, ) Increasing on (0, ) if b > 1 Decreasing on (0, ) if b < 1 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Example A normal child’s systolic blood pressure may be approximated by the function where p(x) is measured in millimeters of mercury, x is measured in pounds, and m and b are constants. Given that m = 19. 4 and b = 18, determine the systolic blood pressure of a child who weighs 92 lb. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Compound Interest Formula 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 © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Example How long will it take an investment of $10, 000 to grow to $15, 000 if it earns an interest at the rate of 12% / year compounded quarterly? Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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. It is about 6. 7% per year. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Present Value Formula for Compound Interest A = The accumulated amount after mt periods P = Principal r = Nominal interest rate/year m = Number of periods/year t = Number of years Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Continuous Compound Interest Formula A = The accumulated amount after t years P = Principal r = Nominal interest rate per year t = Number of years Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Differentiation of Exponential Functions Derivative of Exponential Function Chain Rule for Exponential Functions If f (x) is a differentiable function, then Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Differentiation of Logarithmic Functions Derivative of Exponential Function Chain Rule for Exponential Functions If f (x) is a differentiable function, then Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Logarithmic Differentiation 1. Take the Natural Logarithm on both sides of the equation and use the properties of logarithms to write as a sum of simpler terms. 2. Differentiate both sides of the equation with respect to x. 3. Solve the resulting equation for . Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Exponential Growth/Decay Models A quantity Q whose rate of growth/decay at any time t is directly proportional to the amount present at time t can be modeled by: Growth Decay Q 0 is the initial quantity k is the growth/decay constant Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Example A certain bacteria culture experiences exponential growth. If the bacteria numbered 20 originally and after 4 hours there were 120, find the number of bacteria present after 6 hours. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Learning Curves An exponential function may be applied to certain types of learning processes with the model: C, A, k are positive constants y y=C x Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Example Suppose that the temperature T, in degrees Fahrenheit, of an object after t minutes can be modeled using the following equation: 1. Find the temperature of the object after 5 minutes. 2. Find the time it takes for the temperature of the object to reach 190°. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Example The number of people R, in a small school district who have heard a particular rumor after t days can be modeled by: If 10 people know the rumor after 1 day, find the number who heard it after 6 days. … Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Example The number of soldiers at Fort Mac. Arthur who contracted influenza after t days during a flu epidemic is approximated by the exponential model: If 40 soldiers contracted the flu by day 7, find how many soldiers contracted the flu by day 15. … Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.