2. 6.6 The Natural Base, e Objectives: Evaluate natural exponential and natural logarithmic functions.

3. The natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously. As n becomes very large, the value of approaches the number 2.71828…, this number has been named e

4. The Natural Exponential Function The exponential function with base e, f(x) = e x is called the natural exponential function and e is called the natural base. The function e x is graphed. Notice that the domain is all real numbers The range is all positive numbers.

5. Ex 1. Evaluate f(x) = e x to the nearest thousandth for each value of x below. a. x= 2 e 2 = 7.389 b. x= ½ e 1/2 = 1.649 c. x = -1 e -1 =.368 d. x = 6 e 6 = 403.429 e. x = 1/3 e 1/3 = 1.396 f. x = -2 e -2 =.135

6. Continuous Compounding Formula

7. Ex 2: An investment of $1000 earns an annual interest rate of 7.6%. Compare the final amounts after 8 years for interest compounded quarterly and for interest compounded continuously. Quarterly A = P(1+ r/n) nt A = 1000(1+.076/4) 4*8 A = 1826.31 Continuously A = Pe rt A = 1000e.076 * 8 A = 1836.75

8. Ex 3: Find the value of $500 after 4 years invested at an annual interest rate of 9% compounded continuously. P = 500 t = 4 r =.09 A = 500e.36 = $716.66

9. The Natural Logarithmic Function The natural logarithmic function y = log e x, abbreviated y = In x, is the inverse of the natural exponential function, y = e x. The function y = In x is graphed along with y = e x. y=x y=e x y = Inx

10. Ex 4 Evaluate f(x) = ln x to the nearest thousand for each value of x below. a.x = 2 ln 2 =.693 b. x = ½ In ½ = -.693 c. x = -1 In -1 = undefined d. x = 5 In 5 = 1.609 e. x= 0.85 In.85 = -.163 f. x = 1 In 1 = 0

11. The natural logarithmic function can be used to solve an equation of the form A = Pe rt for the exponent t in order to find the time it takes for an investment that is compounded continuously to reach a specific amount. **** In e = 1 **** Because log e e = 1

12. Ex 5 How long does it take for an investment to double at an annual interest rate of 8.5% compounded continuously? A = Pe rt 2 P = Pe rt 2 = e 0.085t ln2 = ln e 0.085t ln 2 = 0.085t t = ln 2/0.085 t = 8.15

13. Ex 5 How long does it take for an investment to triple at an annual interest rate of 7.2% compounded continuously?

14. ► Ex 7 Radiocarbon Dating Suppose that archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original carbon-14. N(t) = N o e -0.00012t 0.78 N o = N o e -0.00012t 0.78 = e -0.00012t ln 0.78 = -0.00012t -0.00012t = ln 0.78 t = ln 0.78/-0.00012 t = 2070.5

18. Homework Integrated Algebra II- Section 6.6 Level A Honors Algebra II- Section 6.6 Level B