Download The Natural Exponential Function

6.6 The Natural Base, e
Objectives:
Evaluate natural exponential and
natural logarithmic functions.
As n becomes very large, the value of
1 n

11  
n

approaches the number 2.71828…,
named e
The natural base, e, is used to estimate
the ages of artifacts and to calculate
interest that is compounded
continuously.
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= 2
e2 = 7.389
d. x = 6
e6 = 403.429
b. x= ½
e1/2 = 1.649
e. x = 1/3
e1/3 = 1.396
c. x = -1
e-1 = .368
f. x = -2
e-2 = .135
Continuous Compounding Formula
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 = 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
A = 500e.36
= $716.66
r = .09
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=ex
y=x
y = Inx
Ex 4 Evaluate f(x) = ln x to the nearest thousandth
for each value of x below.
a.x = 2
b. x = ½
ln 2 = .693 In ½ = -.693
d. x = 5
In 5 = 1.609
c. x = -1
In -1 = undefined
e. x= 0.85
In.85 = -.163
f. x = 1
In 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
double at an annual interest rate of 8.5%
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 investment
to triple at an annual interest rate of 7.2%
compounded continuously?
► 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) = 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
Homework
Integrated Algebra II- Section 6.6 Level A
Academic Algebra II- Section 6.6 Level B