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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

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 = 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)

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

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

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

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 = 3x

y = (½)x

y = 1x

Jeff Bivin -- LZHS

(0, 1)

Jeff Bivin -- LZHS

x - 3 = 0

x = 3

3

2

(3, -1)

y = -2

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

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

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

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

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

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

- Product Property
- Quotient Property
- Power Property
- Property of Equality

multiplication

addition

multiplication

addition

division

subtraction

division

subtraction

p

logb(mp)

logb(mp) = p•logb(m)

product property

power property

quotient property

product property

power property

quotient property

product property

distributive property

power property

(

(

power property

product property

quotient property

(

(

Power property

group / factor

product property

quotient property

Power property

(

(

re-organize

group

product property

quotient property

Property of Equality

We must check for false solutions!

checks!

Solution

Condense left side

Property of Equality

We must check for false solutions!

checks!

Solution

Condense left side

Convert to exponential form

We must check for false solutions!

fails

The argument must be positive

checks!

Solution

Convert to exponential form

We must check for false solutions!

checks!

checks!

Solution