© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 8-6 Exponential and Logarithmic Functions, Applications, and Models

© 2008 Pearson Addison-Wesley. All rights reserved Exponential and Logarithmic Functions, Applications, and Models Exponential Functions and Applications Logarithmic Functions and Applications Exponential Models in Nature

© 2008 Pearson Addison-Wesley. All rights reserved Exponential Function An exponential function with base b, where b > 0 and is a function of the form where x is any real number.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Exponential Function (b > 1) y x The x-axis is the horizontal asymptote of each graph. x2 x2 x –11/

© 2008 Pearson Addison-Wesley. All rights reserved Example: Exponential Function (0 < b < 1) y x The x-axis is the horizontal asymptote of each graph. x(1/2) x –24 – /2

© 2008 Pearson Addison-Wesley. All rights reserved Graph of 1. The graph always will contain the point (0, 1). 2. When b > 1 the graph will rise from left to right. When 0 < b < 1, the graph will fall from left to right. 3. The x-axis is the horizontal asymptote. 4. The domain is and the range is

© 2008 Pearson Addison-Wesley. All rights reserved Exponential with Base e y x

© 2008 Pearson Addison-Wesley. All rights reserved Compound Interest Formula Suppose that a principal of P dollars is invested at an annual interest rate r (in percent, expressed as a decimal), compounded n times per year. Then the amount A accumulated after t years is given by the formula

© 2008 Pearson Addison-Wesley. All rights reserved Example: Compound Interest Formula Suppose that $2000 dollars is invested at an annual rate of 8%, compounded quarterly. Find the total amount in the account after 6 years if no withdrawals are made. Solution

© 2008 Pearson Addison-Wesley. All rights reserved Example: Compound Interest Formula Solution (continued) There would be $ in the account at the end of six years.

© 2008 Pearson Addison-Wesley. All rights reserved Continuous Compound Interest Formula Suppose that a principal of P dollars is invested at an annual interest rate r (in percent, expressed as a decimal), compounded continuously. Then the amount A accumulated after t years is given by the formula

© 2008 Pearson Addison-Wesley. All rights reserved Example: Continuous Compound Interest Formula Suppose that $2000 dollars is invested at an annual rate of 8%, compounded continuously. Find the total amount in the account after 6 years if no withdrawals are made. Solution

© 2008 Pearson Addison-Wesley. All rights reserved Example: Compound Interest Formula Solution (continued) There would be $ in the account at the end of six years.

© 2008 Pearson Addison-Wesley. All rights reserved Exponential Growth Formula The continuous compound interest formula is an example of an exponential growth function. In situations involving growth or decay of a quantity, the amount present at time t can often be approximated by a function of the form where A 0 represents the amount present at time t = 0, and k is a constant. If k > 0, there is exponential growth; if k < 0, there is exponential decay.

© 2008 Pearson Addison-Wesley. All rights reserved Definition of For b > 0,

© 2008 Pearson Addison-Wesley. All rights reserved Exponential and Logarithmic Equations Exponential Equation Logarithmic Equation

© 2008 Pearson Addison-Wesley. All rights reserved Logarithmic Function A logarithmic function with base b, where b > 0 and is a function of the form

© 2008 Pearson Addison-Wesley. All rights reserved Graph of The graph of y = log b x can be found by interchanging the roles of x and y in the function f (x) = b x. Geometrically, this is accomplished by reflecting the graph of f (x) = b x about the line y = x. The y-axis is called the vertical asymptote of the graph.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Logarithmic Functions y x y x

© 2008 Pearson Addison-Wesley. All rights reserved Graph of 1. The graph always will contain the point (1, 0). 2. When b > 1 the graph will rise from left to right. When 0 < b < 1, the graph will fall from left to right. 3. The y-axis is the vertical asymptote. 4. The domain is and the range is

© 2008 Pearson Addison-Wesley. All rights reserved Natural Logarithmic Function g(x) = ln x, called the natural logarithmic function, is graphed below. y x

© 2008 Pearson Addison-Wesley. All rights reserved Natural Logarithmic Function The expression ln e k is the exponent to which the base e must be raised in order to obtain e k. There is only one such number that will do this, and it is k. Thus for all real numbers k,

© 2008 Pearson Addison-Wesley. All rights reserved Example: Doubling Time Suppose that a certain amount P is invested at an annual rate of 5% compounded continuously. How long will it take for the amount to double (doubling time)? Solution Sub in 2P for A (double). Divide by P.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Doubling Time Solution (continued) Therefore, it would take about 13.9 years for the initial investment P to double. Divide by.05 Take ln of both sides. Simplify.

© 2008 Pearson Addison-Wesley. All rights reserved Exponential Models in Nature Radioactive materials disintegrate according to exponential decay functions. The half- life of a quantity that decays exponentially is the amount of time it takes for any initial amount to decay to half its initial value.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Half-Life Carbon 14 is a radioactive form of carbon that is found in all living plants and animals. After a plant or animal dies, the radiocarbon disintegrates. The amount of carbon 14 present after t years is modeled by the exponential equation a) What is the half-life of carbon 14? b) If an initial sample contains 1 gram of carbon 14, how much will be left in 10,000 years?

© 2008 Pearson Addison-Wesley. All rights reserved Example: Half-Life Solution a) The half-life of carbon 14 is about 5700 years.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Half-Life Solution b) There will be about.30 grams remaining.