Exponential and logarithmic functions
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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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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 and logarithmic functions

Exponential

graphs with

translations


F x 2 x

f(x) = 2x


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


Exponential and logarithmic functions

Exponential

DECAY


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.


A new number1

A new Number


Y 2 x4

y = 3x

y = 2x

Graph y = ex

y = ex


Exponential and logarithmic functions

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 and logarithmic functions

exponential

logarithmic


Evaluate

Evaluate


Evaluate1

Evaluate


Evaluate2

Evaluate


Evaluate3

Evaluate


Evaluate4

Evaluate


Evaluate5

Evaluate


Y 2 x7

y = 2x

x = 2y

INVERSE

y=log2x


Exponential and logarithmic functions

x = 2y

y = log2x

y = log3x

y = log5x


Exponential and logarithmic functions

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

??????


Change of base formula

Change of Base Formula

if b = 10


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


Product property1

Product Property


Quotient property

Quotient Property

division

subtraction

division

subtraction


Quotient property1

Quotient Property


Power property

Power Property

p

logb(mp)

logb(mp) = p•logb(m)


Power property1

Power Property


Property of equality

Property of Equality


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


Solve for x11

Solve for x


Solve for x12

Solve for x


Solve for x13

Solve for x


Solve for x14

Solve for x


Solve for x15

Solve for x


Solve for x16

Solve for x


Solve for x17

Solve for x


Solve for x18

Solve for x


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