Objectives. Use the number e to write and graph exponential functions representing realworld

Transcription

1 Objectives Use the number e to write and graph exponential functions representing realworld situations. Solve equations and problems involving e or natural logarithms.

2 natural logarithm Vocabulary natural logarithmic function

3 Recall the compound interest formula A = P(1 + n ) nt, where A is the amount, P is the principal, r is the annual interest, n is the number of times the interest is compounded per year and t is the time in years. r Suppose that $1 is invested at 100% interest (r = 1) compounded n times for one year as 1 represented by the function f(n) = P(1 + n ) n.

4 As n gets very large, interest is continuously compounded. Examine the graph of f(n)= 1 (1 + n ) n. The function has a horizontal asymptote. As n becomes infinitely large, the value of the function approaches approximately This number is called e. Like π, the constant e is an irrational number.

5 Exponential functions with e as a base have the same properties as the functions you have studied. The graph of f(x) = e x is like other graphs of exponential functions, such as f(x) = 3 x. The domain of f(x) = e x is all real numbers. The range is {y y > 0}.

6 Caution The decimal value of e looks like it repeats: The value is actually There is no repeating portion.

7 Example 1: Graphing Exponential Functions Graph f(x) = e x Make a table. Because e is irrational, the table values are rounded to the nearest tenth. x f(x) = e x

8 Check It Out! Example 1 Graph f(x) = e x 3. Make a table. Because e is irrational, the table values are rounded to the nearest tenth. x f(x) = e x

9 A logarithm with a base of e is called a natural logarithm and is abbreviated as ln (rather than as log e ). Natural logarithms have the same properties as log base 10 and logarithms with other bases. The natural logarithmic function f(x) = ln x is the inverse of the natural exponential function f(x) = e x.

10 The domain of f(x) = ln x is {x x > 0}. The range of f(x) = ln x is all real numbers. All of the properties of logarithms from Lesson 7-4 also apply to natural logarithms.

11 Example 2: Simplifying Expression with e or ln Simplify. A. ln e 0.15t B. e3ln(x +1) ln e 0.15t = 0.15t e 3ln(x +1) = (x + 1) 3 C. ln e 2x + ln e x ln e 2x + ln e x = 2x + x = 3x

12 Check It Out! Example 2 Simplify. a. ln e 3.2 b. e 2lnx ln e 3.2 = 3.2 e 2lnx = x 2 c. ln e x +4y ln e x + 4y = x + 4y

13 The formula for continuously compounded interest is A = Pe rt, where A is the total amount, P is the principal, r is the annual interest rate, and t is the time in years.

14 What is the total amount for an investment of $500 invested at 5.25% for 40 years and compounded continuously? A = Pe rt Example 3: Economics Application A = 500e (40) A Substitute 500 for P, for r, and 40 for t. Use the e x key on a calculator. The total amount is $

15 What is the total amount for an investment of $100 invested at 3.5% for 8 years and compounded continuously? A = Pe rt Check It Out! Example 3 A = 100e 0.035(8) A Substitute 100 for P, for r, and 8 for t. Use the e x key on a calculator. The total amount is $

16 Simplify. Lesson Quiz 1. ln e 10t 0.25 lnt 10t 2. e t ln e x x 4. 2ln e x2 2x 2 5. What is the total amount for an investment of $1000 invested at 7.25% for 15 years and compounded continuously? $