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.