1. 6.6 The Natural Base, e Objectives: Evaluate natural exponential and natural logarithmic functions.

2. The natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously.

3. 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.

4. 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

5. Continuous Compounding Formula

6. 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

7. 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

8. 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

9. Ex 4 Evaluate f(x) = ln x to the nearest thousandth 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

10. 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 ****

11. Ex 5 How long does it take for an investment to double at an annual interest rate of 8.5% compounded continuously?

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

13. ► 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.

16. Homework Pg. 397-398 #12-30 even #72-75