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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

ExponentialandLogarithmic Functions

By: Hendrik Pical to Revition

Exponential and Logarithmic Functions

Last Updated: January 30, 2011


With your graphing calculator graph each of the following
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 2 x

y = 3x

y = 2x


Y 2 x1

y = 3x

y = 5x

y = 2x

y = 1x


Y 2 x2

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



F x 2 x 3
f(x) = 2x-3

x - 3 = 0

x = 3

(3, 1)

3


F x 2 x 2 3
f(x) = 2x+2 - 3

x + 2 = 0

x = -2

3

2

(-2, -2)

y = -3


F x 2 x 4 2

flip

flip

f(x) = -(2)x-4 – 2

x - 4 = 0

x = 4

4

2

y = -2

(4, -3)


Compound interest
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 interest1
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 interest2
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



With your graphing calculator graph each of the following1
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 2 x3

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

(0, 1)

Jeff Bivin -- LZHS


F x x 3 2 2 x 3 2
f(x) = (½)x-3 - 2 = (2)-x+3 - 2

x - 3 = 0

x = 3

3

2

(3, -1)

y = -2


A new number
A new Number

We could use a spreadsheet to determine an approximation.



Y 2 x4

y = 3x

y = 2x

Graph y = ex

y = ex


Graph:

y = ex+2

y = ex+2

y = ex

x + 2 = 0

x = -2


Compound interest continuously
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 continuously1
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
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 growth1
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


Y 2 x5
y = 2x

x = 2y

INVERSE


Y 2 x6
y = 2x

x = 2y

INVERSE

How do we

solve this

exponential

equation

for the variable y?

?


Logarithms
LOGARITHMS

exponential

logarithmic

b > 0

A > 0


exponential

logarithmic








Y 2 x7
y = 2x

x = 2y

INVERSE

y=log2x


x = 2y

y = log2x

y = log3x

y = log5x


x = (½)y

y = log½x


Solve for x
Solve for x

log2(x+5) = 4

24= x + 5

16= x + 5

11= x


Solve for x1
Solve for x

logx(32) = 5

x5= 32

x5 = 25

x = 2


Evaluate6
Evaluate

log3(25) = u

3u= 25

3u = 52

??????



Evaluate7
Evaluate

log3(25)

= 2.930


Evaluate8
Evaluate

log5(568)

= 3.941


Properties of logarithms
Properties of Logarithms

  • Product Property

  • Quotient Property

  • Power Property

  • Property of Equality


Product property
Product Property

multiplication

addition

multiplication

addition



Quotient property
Quotient Property

division

subtraction

division

subtraction



Power property
Power Property

p

logb(mp)

logb(mp) = p•logb(m)




Expand
Expand

product property

power property


Expand1
Expand

quotient property

product property

power property


Expand2
Expand

quotient property

product property

distributive property

power property


Condense
Condense

(

(

power property

product property

quotient property


Condense1
Condense

(

(

Power property

group / factor

product property

quotient property


Condense2
Condense

Power property

(

(

re-organize

group

product property

quotient property


Solve for x2
Solve for x

Property of Equality

We must check for false solutions!


Solve for x3
Solve for x

checks!


Solve for x4
Solve for x

Solution


Solve for n
Solve for n

Condense left side

Property of Equality

We must check for false solutions!


Solve for n1
Solve for n

checks!


Solve for n2
Solve for n

Solution


Solve for x5
Solve for x

Condense left side

Convert to exponential form

We must check for false solutions!


Solve for x6
Solve for x

fails

The argument must be positive

checks!


Solve for x7
Solve for x

Solution


Solve for x8
Solve for x

Convert to exponential form

We must check for false solutions!


Solve for x9
Solve for x

checks!

checks!


Solve for x10
Solve for x

Solution










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