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) = 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.

4. c. x = -1 e-1 = .368 x= 2 e2 = 7.389 b. x= ½ e1/2 = 1.649 d. x = 6
Ex 1. Evaluate f(x) = ex to the nearest
thousandth for each value of x below.
c. x = -1
e-1 = .368
x= 2
e2 = 7.389
b. x= ½
e1/2 = 1.649
d. x = 6
e6 =
e. x = 1/3
e1/3 = 1.396
f. x = -2
e-2 = .135

5. Continuous Compounding Formula

6. Continuously Quarterly A = Pert A = P(1+ R/N)NT A = 1000e .076 * 8
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( /4)4*8
A =
Continuously
A = Pert
A = 1000e .076 * 8
A =

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

8. The Natural Logarithmic Function
The natural logarithmic function y = logx,
appreviated y = In x, is the inverse of the natural
exponential function, y = ex.
The function y = Inx is graphed along with y = ex.
y=ex
y=x
y = Inx

9. b. x = ½ In ½ = -.693 c. x = -1 In -1 = undefined d. x = 5
Ex 4 Evaluate f(x) = lnx to the nearest thousandth
for each value of x below.
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 = 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 ****

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 shows how radiocarbon dating is used to estimate the age of an archaeological artifact.
Ex 7 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.