Math-3 Lesson 8-5

Review Unit 5 (part 1)(Logarithms)

Vocabulary: The “Initial” Value

xxf )3(7)(

The initial value of the function is the coefficient

of the exponential.

xxg )5(3)(

What is the initial value of the following functions ?

xxg 7)(

How much money will be in a savings account

5 years after $300 was deposited if the

interest rate is 6%?

trAtA )1()( 0

Amount of $$

In the account

Initial

deposit

Annual

interest

rate

Years after

the deposit

5)06.01(300)5( A

47.401$)5( A

How much money will there be in an account after

5 years if it earns 6% interest compounded

quarterly if the initial deposit is $300?

kt

krAtA )1()( 0

Amount of $$

In the accountInitial

deposit

Annual

interest rateYears after

the deposit

# of times per year

interest is paid

)5(4)4

06.01(300)5( A

06.404$)5( A

Continuous compounding When they pay the

interest continuously.

rteAtA 0)(

Amount of $$

In the account

Initial

deposit

Annual

interest rate

Years after

the deposit

You can find this

on your calculator

How much money will there be in an account after 5 years

if it earns 6% interest compounded continuously if the initial

deposit is $300?

rteAtA 0)(

Amount of $$

In the account

Initial

deposit

Annual

interest rate

Years after

the deposit

)5(06.0300)5( eA

96.404$)5( A

Transforming Exponential FunctionsDescribe how to transform the graph of:

Into the graph of :

xxf 2)(

42*3)( 2 xxg VSF = 3, right 2, down 4

What is the:

a) Initial value

b) y-intercept

c) x-intercept

d) Horizontal asymptote

e) Domain

f) Range.

a) 3

b) 413

c) 2.415

d) y = -4

e) (-∞, ∞)

f) (-4, ∞)

Exponential Growth and Decay

xabxf )(

For what range of values of ‘b’

will result in exponential growth ?

For what range of values of ‘b’

will result in exponential decay ? 0 < b < 1

b > 1

Your turn: What is the equation?

diff in x

x -2 -1 0 1 2 3 4 5y 2 6 18 54 162 486

diff in y

+1 +1 +1 +1

*3 *3 *3*3

1. ‘f(x)’ is exponential xabxf )(

2. “initial value” = 2 2a xbxf 2)(

3. “growth factor” = 3 3b xxf )3(2)(

4. Check it: f(3) = ?3)3(2)3( f 27*2 54

54)3( f

Comparing Rates of Growth: Which function grows the fastest?

xxf 3)( 2)( xxg

3)( xxh

4)( xxk

]000,1 ,10[y

]100,10[y ]100,10[y ]100,10[y ]100,10[y

]100,10[y ]100,10[y ]000,10 ,10[y

xxf 3)( 2)( xxg 3)( xxh

4)( xxk

xxf 3)(

window

4)( xxk

xxf 3)( 4)( xxk

xxf 3)(

Domain and Range

• The domain of f is the range of f -1

• The range of f is the domain of f -1

xxf log)(

Exponential Function

xxf 10)(

Domain = ?

Range = ?

Domain = ?

Range = ?

(-∞, ∞)

(0, ∞)

Logarithm

Function

Inverse

Functions

(0, ∞)

(-∞, ∞)

Horizontal asymptote = ?

y = 0Vertical asymptote = ?

x = 0

Transformations of the Log Function

xxf log)( )1log()( xxgRight 1 translation

Domain = ?

Range = ?

(1, ∞)

(-∞, ∞)

vertical asymptote is the

left/right shift

x = 0 asymptote = ? x = 1

Transformations of the Log Function

xxf log)(

1)2log(3)( xxgReflected x-axis

VSF = 3

Right 2 translation

Up 1 translation

Domain = ?

Range = ?

(2, ∞)

(-∞, ∞)

asymptote = ? x = 2

This is NOT an exponential

(has a vertical asymptote

NOT a horizontal asymptote).

What is a logarithm?

A logarithm is another way of writing an exponent.

x8log 282 x

Both of these equations are saying the same

thing:

“2 raised to what power is 8?”

x is the exponent Log = exponent

base = logarand

Convert to exponential form

x100log 10

5)27(log3 x

6)42(log3 9 x

10010 x

2735 x

429)

36(

x

exponent

base

exponentlog (logarand) =

4292 x

What does it mean?8log

x8log 810 x

010 x10

1

110

108

8.0x 9.0x

Find on your calculator.8log 903.08log

10 raise to what exponent equals 8?

Finding the Inverse

xxf )5(2)(

?)(1 xf

yx )5(2

yx)5(

2 Base: 5

“A log is an exponent”

2log5

xy

2log)( 5

1 xxf

Finding the Inverse

2)3()( 1 xxf

?)(1 xf

Base: 3

“A log is an exponent”

2log1 3 xy

12log)( 3

1 xxf

2)3( 1 yx

1)3(2 yx

“isolate” the exponential”

12log3 xy

Properties of Logarithms: Log of a Product.

SRRS bbb loglog)(log

5log3log15log 222

Log of a “product”

log of a product = sum of the logs of the factors.

25xx

Logarithm: another way of writing the exponent

Analogous to:

Product of powers

Add the exponents25 x

xy3log yx 33 loglog

“Expanding the Product”

45log 3

5log3log3log 333

5*3*345

5log3log2 33

Properties of Logarithms

3log4 2

52log

2log5

Log of a Power

c

b Rlog

4

2 3log 32log

cRc blog

25x Logarithm: another way of

writing the exponent

2*5x10x

Gotcha’

Which one?y3log5

53log y5log3log y

y3log5 53log y 553log y 53log y

Exponent ‘5’ is applied to ‘y’? or both ‘3’ and ‘y’ ?

Properties of Logarithms Log of a Quotient

SRS

Rbbb logloglog

2

5log 3

2log5log 33 “expand the quotient”

3ln8ln “condense the quotient”

3

8ln

“Negative Log” denominator of the logarand

Expand the Logarithm

Factors of numerator become

positive logs

53

2log

y

x

yx log53log2log

yx log53loglog2log

53log2log yx

Factors of denominator

become negative logs

Change of base formula: can convert any base to any other base

?9log 2 2log

9log

301.0

9542.0 17.3

?9log 2

2ln

9ln

69315.0

19722.2 17.3

BUT, more useful to convert to either (1) base 10, or (2) base ‘e’