session 6 : 9/221 exponential and logarithmic functions
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Session 6 : 9/22 1
Exponential and Logarithmic Functions
Session 6 : 9/22 2
Exponential Functions Definition: If a is some number greater than 0, and
a = 1, then the exponential function with base a is:
xaxf )(Examples:
xx
x
x
x
xf
y
xf
93)(
9
1
9
1
2)(
2
Session 6 : 9/22 3
Properties of Exponents If a and b are positive numbers:
xx
x
xx
xxx
aa
b
a
b
a
baab
1 .7
.6
.5
xyyx
yxy
x
yxyx
aa
aa
a
aaa
a
4.
.3
.2
1 .1 0
Session 6 : 9/22 4
Graphs of Exponential Functions Point plotting or graphing tool
If base is raised to positive x, function is an increasing exponential. If base is raised to negative x, function is a decreasing exponential
If a>1 and to a (+)x : Increasing Exponential If a<1 and to a (+)x : Decreasing Exponential If a>1 and to a (-)x: Decreasing Exponential If a<1 and to a (-)x: Increasing Exponential
Session 6 : 9/22 5
Example exponentials
0
10
20
30
40
50
60
70
-4 -2 0 2 4
x
f(x)
f(x)=2^x
f(x)=3^x
f(x)=1/2^x
f(x)=3 (̂-x)
f(x)=1/4 (̂-x)
Session 6 : 9/22 6
Sketching an Exponential
Find horizontal asymptote and plot several points How do we find horizontal asymptote?
Take the limit as x approaches infinite (for decreasing exponentials) or negative infinite (for increasing exponentials)
Session 6 : 9/22 7
INC
REA
SIN
G E
XPO
NEN
TIA
L
Asymptote for increasing exponential function
Asymptote for decreasing exponential functionx +
x -
8
8
Session 6 : 9/22 8
Natural Exponential Functions In calculus, the most convenient (or natural)
base for an exponential function is the irrational number e (will become more obvious once we start trying to differentiate/integrate…)
e ≈ 2.718
Simplest Natural Exponential:
xexf )(
Session 6 : 9/22 9
Graph of the Natural Exponential Function
Sample Natural Exponential Graphs
0
5
10
15
20
25
30
35
40
45
50
-4 -2 0 2 4
x
y
e x̂
e -̂x
2e x̂
e 2̂x
Session 6 : 9/22 10
Exponential Growth Exponential functions (particularly natural exponentials) are
commonly used to model growth of a quantity or a population What growth is unrestricted, can be described by a form of the
standard exponential function (probably will have multiplying constants, slight changes…):
When growth is restricted, growth may be best described by the logistic growth function:
tetf )(
ktbe
atf
1
)(
Where a, b, and k are constants defined for a given population under specified conditions.
Session 6 : 9/22 11
Comparing Exponential v. Logistic Growth Function
y
x
Exponential
Logistic Growth Function
Session 6 : 9/22 12
Derivatives of The Natural Exponential Function From now on, ‘Exponential Function’ will imply an function with
base e
Previously, we said that e is the most convenient base to use in calculus. Why?
Very simple derivative!
dx
duee
dx
d
eedx
d
uu
xx
][
][
Chain rule, where u is a function of x
Session 6 : 9/22 13
What does this mean graphically?
xexf )(For the function
the slope at any point xis given by the derivative
xexf )(
1
slop
e =
e1
2
slop
e =
e2
slope =
e0 =1
Session 6 : 9/22 14
Examples:
2
2)(
)(
2)( 3
xx
x
x
eexf
xexf
exf
Session 6 : 9/22 15
Logarithmic Functions Review of ‘log’
1000103)1000log(
2552)25(log
823)8(log
3
25
32
If no base specified, log10
Session 6 : 9/22 16
The Natural Log Natural Log=loge=ln
Definition of the natural log: The natural logarithmic function, denoted by ln(x), is defined as:
bx )ln(
xeb
)(log
)(log)(log
xb
xe
e
eb
e
xebx b ifonly and if )ln(
Why?
Session 6 : 9/22 17
Important Properties of Logarithmic Functions
xnx
yxy
x
yxxy
xe
xe
n
x
x
ln)ln( .5
)ln()ln(ln .4
lnln)ln( .3
.2
)ln( .1)ln(
Natural log is inverse of exponentialExponential is inverse of natural log
Session 6 : 9/22 18
Examples: Solve the following logarithmic functions for x
Simplify the following:
4)ln(
3)ln(
2
x
x
)2ln(
)8ln(
3)ln(
xe
e
e
Session 6 : 9/22 19
Examples: Solving Exponential and Logarithmic Equations
7)3ln(5
4)ln(
92
8
5.0
x
x
e
e
t
x
Session 6 : 9/22 20
Example: Doubling Time For an account with initial balance P, the function
for the account balance (A) after t years (with annual interest rate r compounded continuously) is given by:
rtPetA )(
Find an expression for the time at which the account balance has doubled.
Session 6 : 9/22 21
Derivative of logarithmic functions:
dx
du
uu
dx
d
xx
dx
d
1][ln
1][ln
Where u is a function of x
Session 6 : 9/22 22
Examples
2ln2)(
ln)(
)23ln()(
:
2
xxf
xxxf
xxxf
ateDifferenti
Session 6 : 9/22 23
Exponential Growth and Decay Law of exponential growth and decay:
ktCey
If y is a positive quantity whose rate of change with respect to time is proportional to the quantity present at any time t, then y is described by:
Where C is the initial valuek is the constant of proportionality (often rate constant)
If k > 0: Exponential GrowthIf k < 0: Exponential Decay
Session 6 : 9/22 24
Example: Modeling population growth:
A researcher is trying to develop an equation to describe bacterial growth, and knows that it will follow the fundamental equation for exponential growth. The following data is available:
At t=2 hours, there are 1x106 cells At t=8 hours, there are 5x108 cells
Write an equation for the exponential growth of bacterial cells by the following steps:
1. Find k 2. Find C using the solution for k 3. Write the full model by plugging in C and k values.
Find the time at which the population is double that of the initial population.
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