Key Terms: exponential function, exponential equation, compound interest, future value, present value, compound amount, continuous compounding.

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1 4.2 Exponential Functions Exponents and Properties Exponential Functions Exponential Equations Compound Interest The Number e and Continuous Compounding Exponential Models Section 4.3 Logarithmic Functions 4-5 Key Terms: exponential function, exponential equation, compound interest, future value, present value, compound amount, continuous compounding Exponents and Properties EXAMPLE 1 Evaluating an Exponential Expression If 4 x f x, find each of the following. Round to the hundredths when necessary. f (b) 5 (a) ( 2) f (c) 2 f 3 (d) f (2.15) Exponential Functions Exponential Function If a 0 and 1 a, then x f x a defines the exponential function with base a. We do not allow as the base for an exponential function.

2 4-6 Chapter 4 Inverse, Exponential, and Logarithmic Functions Characteristics of the Graph of f ( x ) = a 1. The points 1,, (0, ), and (1, ) are on the graph. x 2. If a 1, then f is an function. If 0 a 1, then f is a function. 3. The is a horizontal asymptote. 4. The domain is and the range is.

3 EXAMPLE 2 Graphing an Exponential Function Graph f x x 1 2. Section 4.3 Logarithmic Functions 4-7 EXAMPLE 3 Graphing Reflections and Translations Graph each function. Give the domain and range. (a) f x 3 x 2 (b) f x 3 x x2 (c) f x 3 2 Solving Exponential Equations Example 4) Solve 1 5 x. Example 5) Solve 125 x1 x3 3 9.

4 4-8 Chapter 4 Inverse, Exponential, and Logarithmic Functions EXAMPLE 6 Solving an Equation with a Fractional Exponent Solve 5/2 x 243. Compound Interest The formula for compound interest (interest paid on both principal and interest) is an important application of exponential functions. Compound Interest If P dollars are deposited in an account paying an annual rate of interest r compounded (paid) n times per year, then after t years the account will contain A dollars, according to the following formula. r A P1 n EXAMPLE 7 Using the Compound Interest Formula Suppose $2500 is deposited in an account paying 6% interest per year compounded semiannually (twice per year). (a) Find the amount in the account after 10 yr with no withdrawals. tn (b) How much interest is earned over the 10-yr period? EXAMPLE 8 Finding Present Value: Brendan Berger must pay a lump sum of $15,000 in 8 yr. (a) What amount deposited today (present value) at 4.8% compounded annually will give $15,000 in 8 yr? (b) If only $10,000 is available to deposit now, what annual interest rate is necessary for the money to increase to $15,000?

5 Section 4.3 Logarithmic Functions 4-9 The Number e and Continuous Compounding The more often interest is compounded within a given time period, the more interest will be earned. Surprisingly, however, there is a limit on the amount of interest, no matter how often it is compounded. Suppose that $1 is invested at 100% interest per year, compounded n times per year. Thus, r A P1 n nt n 1 becomes A 1. What happens to this expression as n increases? n n 1 A 1 n n (rounded) ,000 1,000,000 Value of e = Continuous Compounding If P dollars are deposited at a rate of interest r compounded continuously for t years, the compound amount A in dollars on deposit is given by the following formula. A Pe EXAMPLE 9 Solving a Continuous Compounding Problem Suppose $8000 is deposited in an account paying 5% interest compounded continuously for 6 yr. Find the total amount on deposit at the end of 6 yr. rt

6 4-10 Chapter 4 Inverse, Exponential, and Logarithmic Functions EXAMPLE 10 Comparing Interest Earned as Compounding is More Frequent In Example 7, we found the total amount on deposit after 10 yr in an account paying 6% semiannually in which $2500 was invested. Find the amounts from the same investment for interest compounded quarterly, monthly, daily, and continuously. Exponential Models EXAMPLE 11 Using Data to Model Exponential Growth Refer to the model given in Example 11. Data from recent past years indicate that future amounts of x carbon dioxide in the atmosphere may grow according to the function y e. Here, amounts are given in parts per million, and x is the year where 1990 x (a) What will be the atmospheric carbon dioxide level in parts per million in 2015? (b) Use a graph of this model to estimate when the carbon dioxide level will be double the level that it was in 2000.

3.1 Exponential Functions and Their Graphs Date: Exponential Function

3.1 Exponential Functions and Their Graphs Date: Exponential Function Exponential Function: A function of the form f(x) = b x, where the b is a positive constant other than, and the exponent, x, is a variable.