Exponential and Logarithmic Functions - PowerPoint PPT Presentation

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Exponential and Logarithmic Functions. By: Hendrik Pical to Revition Exponential and Logarithmic Functions. Last Updated: January 30, 2011. With your Graphing Calculator graph each of the following. y = 2 x. y = 3 x. y = 5 x. y = 1 x.

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Exponential and Logarithmic Functions

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ExponentialandLogarithmic Functions

By: Hendrik Pical to Revition

Exponential and Logarithmic Functions

Last Updated: January 30, 2011

With your Graphing Calculatorgraph each of the following

y = 2x

y = 3x

y = 5x

y = 1x

Determine what is happening when the base is changing in each of these graphs.

y = 3x

y = 3x

y = 5x

y = 1x

y = 3x

y = 5x

y = 2x

y = 4x

y = 10x

y = (3/2)x

Determine where each of the following would lie?

y=10x

y=4x

y = (3/2)x

y = 1x

Exponential

graphs with

translations

x - 3 = 0

x = 3

(3, 1)

3

x + 2 = 0

x = -2

3

2

(-2, -2)

y = -3

flip

flip

x - 4 = 0

x = 4

4

2

y = -2

(4, -3)

Compound Interest

You deposit \$5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly?

A = Final amount = unknown

P = Principal = \$5000

r = rate of interest = .045

n = number of times compounded per year = 4

t = number of years compounded = 10

Compound Interest

You deposit \$5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly?

A = unknown

P = \$5000

r = .045

n = 4

t = 10

Compound Interest

You deposit \$5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly?

weekly?

A = unknown

P = \$5000

r = .045

52

n = 4

t = 10

Exponential

DECAY

With your Graphing Calculatorgraph each of the following

y = (1/2)x

y = (1/3)x

y = 1x

Determine what is happening when the base is changing in each of these graphs.

y = (1/3)x

y = 5x

y = 2x

y = 3x

y = (½)x

y = 1x

Jeff Bivin -- LZHS

f(x) = 2-x = (1/2)x

(0, 1)

Jeff Bivin -- LZHS

x - 3 = 0

x = 3

3

2

(3, -1)

y = -2

A new Number

We could use a spreadsheet to determine an approximation.

y = 3x

Graph y = ex

y = ex

Graph:

y = ex+2

y = ex+2

y = ex

x + 2 = 0

x = -2

Compound Interest-continuously

You deposit \$5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded continuously?

A = Final amount = unknown

P = Principal = \$5000

r = rate of interest = .045

t = number of years compounded = 10

Compound Interest-continuously

You deposit \$5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded continuously?

A = unknown

P = \$5000

r = .045

t = 10

Bacteria Growth

You have 150 bacteria in a dish. It the constant of growth is 1.567 when t is measured in hours. How many bacteria will you have in 7 hours?

y = Final amount = unknown

n = initial amount = 150

k = constant of growth = 1.567

t = time = 7

Bacteria Growth

You have 150 bacteria in a dish. It the constant of growth is 1.567 when t is measured in hours. How many bacteria will you have in 7 hours?

y = unknown

n = 150

k = 1.567

t = 7

x = 2y

INVERSE

y = 2x

x = 2y

INVERSE

How do we

solve this

exponential

equation

for the variable y?

?

exponential

logarithmic

b > 0

A > 0

exponential

logarithmic

x = 2y

INVERSE

y=log2x

x = 2y

y = log2x

y = log3x

y = log5x

x = (½)y

y = log½x

log2(x+5) = 4

24= x + 5

16= x + 5

11= x

logx(32) = 5

x5= 32

x5 = 25

x = 2

log3(25) = u

3u= 25

3u = 52

??????

if b = 10

log3(25)

= 2.930

log5(568)

= 3.941

Properties of Logarithms

• Product Property

• Quotient Property

• Power Property

• Property of Equality

multiplication

multiplication

division

subtraction

division

subtraction

Power Property

p

logb(mp)

logb(mp) = p•logb(m)

product property

power property

Expand

quotient property

product property

power property

Expand

quotient property

product property

distributive property

power property

Condense

(

(

power property

product property

quotient property

Condense

(

(

Power property

group / factor

product property

quotient property

Condense

Power property

(

(

re-organize

group

product property

quotient property

Solve for x

Property of Equality

We must check for false solutions!

checks!

Solution

Solve for n

Condense left side

Property of Equality

We must check for false solutions!

checks!

Solution

Solve for x

Condense left side

Convert to exponential form

We must check for false solutions!

Solve for x

fails

The argument must be positive

checks!

Solution

Solve for x

Convert to exponential form

We must check for false solutions!

checks!

checks!

Solution