Math‐3 

Unit 5 Review(Logs and Exponentials and their 

Applications)

What is the equation that passes through the data?

x y

1

2

3

4

5

0.250.5124

xABxf )(31 AB42 AB

3

4

12

ABAB

B23)2(1 A

81

A

xxf )2(81)(

xxf )2(21)( 3

xxf )2(2)( 332)( xxf

The graph of can be obtainedfrom by reflecting it acrossthe y-axis.

Transforming Exponential Functions

xxf 2)(xxf 2)(

xxf 2)(xxf 2)( xxxf

212)(

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:Identify the following as either “growth” or “decay”.

xxf 2)(

xxf )3()(

xxf 31)(

xxf 234)(

dabxf cx ))(1()1()(If negative:

Reflect across x-axis

Initial value:Crosses y-axis at

y = a + dGrowth factor:

If negative:Reflect across y-axis

Horizontal shift

vertical shift and horizontal Asymptote

xxf 2)3(4)( )2(1)3(4 x

Finding the Inverse: exchange the locations of ‘x’ and ‘y’ in the equation then solve for ‘y’.

2)2()( xxf2)2( xy2)2( yx

2)2( yx

2 yx

yx 2

xy 2

xxf log)(

Exponential Function

xxf 10)( Domain = ?Range = ?

Domain = ?Range = ?

(-∞, ∞)(0, ∞)

LogarithmFunction

InverseFunctions

(0, ∞)(-∞, ∞)

Horizontal asymptote = ? y = 0 Vertical

asymptote = ?x = 0

Transformations of the Log Functionxxf log)(

1)2log(3)( xxgReflected x-axisVSF = 3Right 2 translationUp 1 translationDomain = ?Range = ?

(2, ∞)(-∞, ∞)

asymptote = ? x = 2NOT exponential (has a vertical asymptote NOT a horizontal asymptote.

Exponential        LogarithmForm             Form

x25log5

x64log4

x81log9

x1000log10

255 x

644 x

819 x

100010 x

Log =

Your Turn:  

What is the base?

x5lnx20log

x8log2

Finding the Inverse

2)3()( 1 xxf

?)(1 xf

Base: 3“A log is an exponent”

2log1 3 xy

12log)( 31 xxf

2)3( 1 yx1)3(2 yx

“isolate” the exponential”

12log3 xy

xy3log yx 33 loglog

Log of a Product Property

45log3

5log3log3log 333

5*3*345

5log3log2 33

“expand the product”

xlog5log “Condense the Product” x5log

Log of a Quotient Property

SRSR

bbb logloglog

25log3 2log5log 33 “expand the quotient”

3ln8ln “condense the quotient”38ln

3log4 2

2log5

Log of a Power Property

cb Rlog

42 3log

32log

c Rc blog

52log

“expand power”

“condense”

Your turn:

Solve: 353 xx

“Isolate the radical”

“Undo the radical”

22)3(53 xx square both sides

9653 2 xxx Get into standard form!!!!!

2 ,7x

1490 2 xx)2)(7(0 xx

3)7(5)7(3

416 3)2(5)2(3

11

xx 41312 77

In some books, this is a property.xx 41312

If the base of exponentials are the same then the exponents equal each other.

1316 x

126 x

+4x +4x

-1 -1

2x

Solving using “convert to same base”

“convert to same base”12 279 xx

1322 33

xx

)1(32*2 33 xx Power of a power Exponent Property

334 33 xx

334 xx

3x

If you don’t recognize: “convert to same base”Use Log of a Power Property12 279 xx

12 27ln9ln xx

27ln)1(9ln2 xx

3x

127ln

9ln2 xx

127ln9ln2 xx

1)6.0(2 xx

1322

xx

334 xx

1275 xx

127ln5ln xx

85.0x

7ln)12(5ln xx

5ln7ln)12( xx

÷ ln 5 ÷ ln 5

21.142.2 xx

xx 42.221.1

x42.121.1

Cannot rewrite the exponentials with the same bases.

)21.1)(12( xx

+1.21 +1.21

-x -x

÷ 1.42 ÷ 1.42

Solve using “undo the exponential”

“Isolate the exponential”

“Undo the exponential”

753 12 x

-5 -5

Change of baseformula

23 12 x

122log3 x

123ln2ln

x

+1 +1÷2 ÷2

815.0x

1263093.0 xx263093.1

Some would say this is a property.“same based log equals same based log”)5(log)74(log 55 xx

Logarands equal each other.4x - 7 = x + 5

3x = 12

x = 4

Subtract ‘x’ from both sides.

Divide both sides by 3

Plug back in to check!)54(log)74*4(log 55

9log9log 55 Checks

Solving Logarithmic Equations“Isolate the logarithm”3)15(log4 x “convert to exponential”

1543 x

5x - 1 = 64

655 xx = 13 Plug back in to check!

Checks

3)113*5(log4

364log4

Solving Logs requiring condensing the product. 

“Isolate the logarithm”2)5log(2log xx“undo the logarithm”

2)5(2log 1010 10 xx

“condense the product”

2x(x - 5) = 100

Inverse of log base 10 is exponent base 10

Quadratic put in standard form

Divide both sides by ‘2’factor

2)5(2log xx

0100102 2 xx05052 xx0)5)(10( xx

x = 10, -5 Zero factor property

x = 10x ≠ -5

We can rewrite the base of any exponential as a power of ‘e’.

693.0k

xy 2

2lnk2ke

xkey kxey

xey 693.0

xy 4 xe 386.1xy 1.1

xy 01.1xy 85.0xy 25.0

xe 095.0xe 010.0

xe 163.0xe 386.1

How can you tell if a base ‘e’ exponential is growth or decay?

kxey Growth: k > 0

Decay: k < 0

Newton’s Law of CoolingA high temperature item will cool off in a lower temperature medium in which it is placed. This cooling off process can be modeled by the following equation.

ktmom eTTTtT )()(

Temperature (as a function of time)

Temperature ofthe medium

Initial Tempof the object

Constant, determinedby the heat transfercharacteristics of the materi

Time

3070)( 11.0 tetT

A cup of hot water is taken out of the microwave oven. Its initial temperature is 100 C.It is placed on the counter in a room whose temperature is30 C. In 5 minutes it has cooledto 72 C. When will it reach 40 C.

1. Draw a graph that shows temperature as a function of time.

2. What is the equation of the graph? (use the following equation). kABtT t )( 3. Convert base “B” to base “e”

then rewrite your equation.

30)9.0(70)( ttT4. T(t) = 40 C. Substitute and solve for ‘t’. min 7.17t

The front row of a rock concert has a sound intensity of

What is the sound level in decibels on the front row of the rock concert?

21 er watts/met10I

1210log10)(

IIL

12

1

1010log10

L 1110log10 10log110dB 110

The loudness of an ambulance was measured to be 120 dB.

What is the sound intensity? (in w/m^2)

1210log10120

I1210

log12 I 12

12

1010

I

01212 101010 I

AciditypH = - log [H+]

The hydrogen-ion concentration of a solution is mole/li 107.5 11

What is the pH of the solution?

pH = - log [ ] 11107.5

pH = 10.3

You deposit $100 money into an account that pays 3.5% interest per year. The interest is “compounded” monthly. How much money will be in the account at the end of the 5th year?

kt

krAtA )1()( 0

)5(12)12035.01(100)5( A

09.119$)5( A

A bank compounds interest continuously. The annual interest rate is 5.5%. How long would it take for the money in the account to triple?

rteAtA 0)( teAA 055.0

003 te 055.03

t055.03ln yrs 97.19t

The “half life” of Carbon-14 (a radioactive isotope of carbon), is 5730 years. Calculate the decay rate for carbon-14.

kteAtA 0)( )5730(

005.0 keAA )(57305.0 ke

k57305.0ln 00012.0k