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,
is called the natural exponential
function and e is called the natural base.
The function 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 =
TRY THIS: top of 394

7. 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. Page 396 #7

9. The Natural Logarithmic Function
The 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=x
y=ex
y = In x

10. What are the domain and range the exponential and natural log functions?
All positive real numbers
All real numbers
All real numbers
All positive real numbers

11. b. x = ½ In ½ = -.693 c. x = -1 In -1 = undefined d. x = 5
Ex 3 Evaluate f(x) = ln x 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

12. 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.

13. Ex. 4 How long does it take for an investment to double at an annual interest rate of 8.5% compounded continuously?

14. “Try This” – p. 395

15. “Critical Thinking” – page 395 Think, Pair, Share

16. p. 403 has a useful summary table:

17. Try these: p. 397 #13-27 odd

18. p. 397: #41-47 odd

19. Homework:
Practice 6.6
Chapter 6 Test – Thu 4/29