6.6 The Natural Base, e6.6 The Natural Base, e

Objectives:Evaluate natural exponential and natural logarithmic functions.

n

n

111

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…, named e

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

Ex 1. Evaluate f(x) = ex to the nearest thousandth for each value of x below.

a. x= 2e2 = 7.389

b. x= ½e1/2 = 1.649

c. x = -1e-1 = .368

d. x = 6e6 = 403.429

e. x = 1/3 e1/3 = 1.396

f. x = -2e-2 = .135

Continuous Compounding Formula

Ex 2: An investment of $1000 earns an annual interest rate of 7.6%. Compare the final amountsafter 8 years for interest compounded quarterly and for interest compounded continuously.

QuarterlyA = P(1+ r/n)nt

A = 1000(1+ .076/4)4*8

A = 1826.31

ContinuouslyA = Pert

A = 1000e .076 * 8

A = 1836.75

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

The Natural Logarithmic FunctionThe natural logarithmic function y = loge x, abbreviated y = In x, is the inverse of the natural exponential function, y = ex. The function y = In x is graphed along with y = ex.

y=xy=ex

y = Inx

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 = ½In ½ = -.693

c. x = -1In -1 = undefined

d. x = 5In 5 = 1.609

e. x= 0.85In.85 = -.163

f. x = 1In 1 = 0

The natural logarithmic function can be used to solve an equation of the form A = Pert 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 ****

Ex 5 How long does it take for an investment to Ex 5 How long does it take for an investment to double at an annual interest rate of 8.5% double at an annual interest rate of 8.5%

compounded continuously? compounded continuously?

A = Pert

2 P = Pert

2 = e0.085t

ln2 = ln e0.085t

ln 2 = 0.085t

t = ln 2/0.085

t = 8.15

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

►Ex 7 Radiocarbon DatingEx 7 Radiocarbon Dating

Suppose that archaeologists find scrolls Suppose that archaeologists find scrolls and claim that they are 2000 years and claim that they are 2000 years old. Tests indicate that the scrolls old. Tests indicate that the scrolls contain 78% of their original carbon-contain 78% of their original carbon-14.14. N(t) = Noe-0.00012t

0.78 No = Noe-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

HomeworkPg. 397-398 #12-30 even #72-75