Unit 8: ExPonential! Logs · • For exponential growth y=abx, with b>1 the value b is the growth...

Unit 8 ExPonential Logs Carli Castellano Caroline Chivily Julia Ashley Chapter 7 Exponential and logarithmic Functjons Section 71 Exploring Exponential Models

Definitions

bull An exponential function is a function with the general form y= abxa=O with bgtO and b does not = 1 In an exponential function the base b is a constant The exponent x is the independent variable with domain the set of real numbers

bull For exponential growthas the value x increases the value of y increases bull For exponential decaYas the value of x increases the value of y decreases approaching

zero bull An asymptote is a line that a graph approaches as x or y increases in absolute value

bull For exponential growth y=abx with bgt1 the value b is the growth factor bull For exponential decay 0ltblt1 and b is the decay factor

Key Concepts bull if agtO and bgt1 the function represents exponential growth bull if agt 0 and 0ltblt1 the function represents exponential decay bull in either case the y-intercept is (Oa) the domain is all real numbers the asymptote is

y=O and the range is ygtO

bull You can model exponential growth or decay with this function- A(t)=a(1+r)t bull For growth or decay to be exponentiala quantity changes by a fixed percentage each

time period

Example 1 Graphing an Exponential function

What is the graph of y=2x

Step 1 Make a table of values

x 2x Y

-4 2 116=00625 0 2deg 1

-3 2-3 18= 0125 1 21 2

-2 2-2 14= 025 2 22 4

-1 2-1 12= 05 3 23 8

Example 2 Identifying Exponential Growth and Decay

Without graphing determine whether the function represents exponential growth or decay Then find the y~intercept

f(x)= 2(O65X) -exponential decay - Y-int (02)

y=3(4X)

- exponential growth - (011)

Example 3 Modeling Exponential Growth Suppose you invest $500 in a savings account that pays 35 annual interest How much will be in the account after five years

a=500 b=(1 +0035)= 1035

y=500(1035)t _ y=500(1035)5 = $59384

Example 5 Writing an Exponential Function The table shows the world population of Iberian lynx in 2003 and 2004 If the trend continues and the population is decreasing exponentially how many Iberian lynx will there be in 2020

Year Population

2003 150

2004 120

y=abX=a( 1 +r)X

1) Define the variables x= number of years since 2003 y=population of lynx

2) Solve for a and b Use y=abx

150=abo - a= 150 120= ab1

_ 120=150b - b=08

3) Find the value of r 1+r=b _1+ r= 08 - r= -02

4) Write the model y=150(08)X

5) Use the model to find the world lynx population in 2020 x=2020-2003=17 y=150(08)17

y= 337 - 3 lynx will be left

Section 72 Properties of Exponential Functions

Essential Understanding The factor a in y=abxcan stretch or compress and possibly reflect the graph of the parent function y=bx

Concept Summary

bull Parent function bull stretch (a gt1)

compression (shrink) (0lt a lt1) Reflection (a lt0) in x-axis

y=b(X-h) +kbull Translations (horizontal by h vertical by k) y= ab(x-h) +kbull All transformations combined

Definitions

Natural base exponential are exponential functions with base e These functions are useful for describing continuous growth or decay Exponential functions with base e have the same properties as other exponential functions

Example 1 Graphing y=aJY How does the graph of y= -13 times 3xcompare to the graph of the parent function


x y=3x y=-13(3X )

-2 19 -127

-1 113 -19

0 1 -13

1 3 -1

2 9 -3

The - 113 in y= -13(JX) reflects the graph of the parent function y=3x across the x-axis and compresses it by the factor 13 The domain and asymptote remain unchanged The y-intercept becomes -13 and the range becomes -13 and the range becomes yltO

Example 2 Translating the Parent Function y=bx

How does the graph of each function compare to the graph of the parent function y= 2(x-4)


x y=2x x y=2x

-2 14 1 2

-1 12 2 4

0 1 3 8

The (x-4) in y=2(x-4) translates the graph of y=2x to the right 4 units The asymptote remains y=O

The y-intercept becomes 116

Example 3 Using an Exponential Model

Time (Min) Temp(OF) Temp - Room Temp (OF)

0 203 135

5 177 109

10 153 85

15 137 69

20 121 53

25 111 43

30 104 36

(Find an exponential function to model the data Use the list feature on the graphing calculator Assume that room temperature is 68deg)

y=1345(0956Y + 68

The initial temperature of a coffee cup of coffee is 203degF An exponential model for the temperature y of the coffee after x minutes is y=1345(0956Y + 68 How long does it take for the coffee to reach a temperature of 100 OF (Enter your function into your calculator and go to your table and locate 100 (or closest number to 100) in the y column (Go to tableset and and 4Tbl if necessary))

IN 391 MINUTES THE COFFEE WILL BE COOLED TO 1000 F

Section 73 Logarithmic Functions as Inverses

bull Recall x=bY

logbx= y

bull The inverse of an exponential function is called a logarithmic function bull For a logarithm y is the exponent that you need to raise b to get x

Example 1 Writing Exponential Equations in Logarithmic Form

1 100=102

log10100=2

2 81=34

log381=4

3) 36=62

log636=2

Example 2Evaluating a Logarithm 1) log832

8Y=32

23Y=25

3y=5 y=53

2) log5125

5Y=125 5Y=53

y=3

3 log432

4Y=32 22Y=25

2y=5 y=52

Example 3 Using a Logarithmic Scale In December 2004 an earthquake with magnitude 93 on the Richter scale hit off the northwest coast of Sumatra The diagram shows the magnitude of an earthquake that hit Sumatra in March 2005 The formula log 112=M1-M2 compares the intensity levels of earthquakes where I is the

intensity level determined by a seismograph and M is the magnitude on a Richter scale How many times more intense was the December earthquake than the March earthquake

log 112=M1-M2 (Use the formula)

log 112=93-87 (Substitue M1=93 and M2=87)

log 112=06 (Simplify)

112=10deg6 (Apply the definition of common logarithm)

=4 (Use your calculator)

A logarithmic function is the inverse of an exponential function

Recall We can graph the inverse of a function by switching the x and y coordinates

Example 4 Graphing a Logarithmic Function What is the graph of y=log4x Describe the domain range y-intercept and asymptotes

4x=y is the inverse Log Domain xgtO

Range all reals y-intercept none VA none

Example 5 Translating y=logtf

How does the graph of the function compare to the parent function 1) y=log2(X-3)+4

right 3 up 4 Domain xgt3 Range All reals

2) y=510g2x

stretch by 5 Domain xgtO Range All reals

3)y=log2(x+4)

left 4 Domainxgt-4 Range All reals

Section 74 Properties of logarithms am X an = am+n(product property) aman = am-n(quotient property) (amt = amn (power property)

Let x=109bm and y=109bn

m=bX

mn=bx bull bY

mn=bx+y

109bmn=109bm+109bn

Product Property 109bmn=109bm+109bn

Quotient Propery 109bmn=109bm-109bn

Power Property 109bmn=n 109bm

Example 1 Simplifying Logarithms Make each a sin91e 109arthim 1) 10945x+10943x

109415x2 (With product power multiply 5x and 3x to get 15x2)

2 109415x (move the 2 to the front with power property)

2) 2 10946-10949

109462-10949 (power property moves the 2 to the front)

109436-10949 (simplify 62to 36)

1094369 (quotient property to divide)

10944 (simplify)

4Y=4 (solve for y) y=1

3) 109432-10942

1094322 (quotient property of 109arithms)

109416 (divide)

109442 (write 16 as a power of 4)

2 (simplify)

Example 2 Expanding Logarithms 1) 109325037

log3250-log337(quotient property)

log32(125)-log337 (product property)

log32+log3125-log337 (product property)

log32+log353-log337 (power property)

log32+310g35-log337 (power property)

2) log39x5

IOg39+log3X5 (product property)

log39+510g3x (power property)

Change of base 10gbm=logmlogb

Example 3 Using the Change of Base Formula 1) log632

log328 16

2) log418

log184 2085

Example 4 Using a Logarithmic Scale

The pH of a substance equals -Iog[H+] where [H-] is the concentration of hydrogen ions Suppose the hydrogen ion concentration for Substance A is twice that for Substance B Which substance has a greater pH level What is the greater pH level minus the lesser pH level Explain

Substance A 2-[H+e]

Substance B [H+e]

A pH=-log[2([H+e])]

B pH=-log[H+e]

pH of A =-(Iog 2+ log [H+eD pH of A =-log2-log[H+ B]

pH of A =-log2+pH of B pH of A =pH of B-Iog 2

pH of B is bigger

Section 75 Exponential and Logarithmic Equations Example 1 Solving an Exponential Equations- Common Base

1) 273X= 81 333X=34

3439x= 9x=4 (take the exponents) x=49

2) 53X=1125 53X= 5-3 (53= 125 5-3= 11125) 3x=-3 x= -1

bull When bases are not the same you can solve an exponential equation by taking the logarithm of each side

Example 2 Solve an Exponential Equation

1) 52x= 130 log52x= log 130 (Make a log with a common base of 10)

2xlog5= log 130 (Power Property moves the 2x) 2x= log13010g5 (Isolate the 2x by dividing log5 on both sides) x= (log1301l0g5)2 (Isolate x by dividing by 2 on both sides) x= 1512

2) 8 + 1OX= 1008 10x= 1000 (Isolate 10xby subtracting 8 from both sides)

log1OX= log1000 (Make a log with a common base of 10) xlog10= log1000 (Power Property moves the x) x= log10001l0g10 (Isolate x by dividing log10 on both sides) x=3

Example 3 Solving an Exponential Equation with a Graph or Table

1) 47x= 250 log447x= log4250 (Put log4 to cancel out the 4 and isolate the 7x)

7x= log25010g4 (Change of base) x= (log2501l0g4)7 (Divide 7 on both sides to isolate the x) x= 056898

Graphically y= 47X

y=250 (Find the pOint of Intersection using your graphing calculator)

Point of Intersection (056898 250)

2) 6x= 4565 log66x= log64565 (Put log6 to cancel out the 6 and isolate the x)

x= log45651og6 (Change of base) x= 47027

Graphically

y=6x

y=4565 (Find the point of intersection using your graphing calculator) Point of Intersection (47027 4565)

Example 4 Modeling with an Exponential Equation Wood is a sustainable renewable natural resource when you manage forests properly Your lumber company has 1200000 trees You plan to harvest 7 of the trees each year how many years will it take to harvest half of the trees

T(n)= a(b)n T(n)=600000 a=1200000 b= 1+ -007=093

T(n)=1200000(093t 600000=1200000(093)n

05=(093) (Divide both sides by 1200000 to isolate term with n)

logo9305= logo93(O93)n (Put logo93 to cancel out 093 and isolate n)

logO5logO93= n n= 955

Example 5 Solving a Logarithmic Equation REMEMBER bY=x

1) log2x=-1

10-1= 2x (Put into exponential form)

10-12 = x (Divide by 2 on both sides to isolate x) x= 005

2) log(3x+1)= 2 102= 3x+1 (Put into exponential form) 99= 3x (Subtract 1 from both sides to isolate the 3x) x= 33 (Divide by 3 on both sides to isolate the x)

Example 6 Using Logarithmic Properties 1) logx- log3= 8

logxl3= 8 (Quotient Property)

108= xl3 (Put in Exponential Form) x= 300000000 (Multiply 3 on both sides)

2) log2x+ logx= 11 log2x(x)= 11 (Product Property) log2x2= 11 (Simplify)

1011 = 2X2 (Put into Exponential Form)

10112= x2 (Divide 2 on both sides)

x2= 50000000000

x= 2236068 (Square Root)

Section 76 Natural Logarithms

bull The inverse or logarithmic of the exponential function y=ex is 10geY=x -+ logex=y

bull Natural logarithmic function is y=lnx

bull Use the same properties as logarithms

Example 1 Simplifying a Natural Logarithmic Expression 1) 31n 5

In 53 (Power Property)

In125

2) In 9 + In 2 In 9(2) (Product Property)

In 18

Example 2 Solving a Natural Logarithmic Equation 1) In x= 2

e2= x (Exponential Form)

x= e2

Check Answer

In e2 = 2

2=2

2) In 2x + In 3 = 2 In 2x(3) = 2 (Product Property)

In 6x= 2

e2= 6x (Put in Exponential Form)

x= e26 (Divide 6 on both sides to isolate x)

Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2

Example 3 Solving an Exponential Equation 1) eX=18

In eX= In 18 (In cancels out the e and isolates the x) x=ln18

2) e2x= 12

In e2x = In 12 (In cancels out the e and isolates the 2x) 2x = In 12 (Divide both sides by 2 to isolate the x)

x= (In 12)2

Example 4 Using Natural Logarithms A spacecraft can attain a stable orbit 300 km above Earth if it reaches a velocity of 77 kms The formula for a rockets maximum velocity v in kilometers per second is v= -00098l + c In R The booster rocket fires for l seconds and the velocity of the exhaust is c kms The ratio of the mass of the rocket filled with fuel to its mass without fuel is R Suppose the rocket shown in the photo has a mass ratio of 25 a firing time of 100 s and an exhaust velocity as shown Can the spacecraft attain a stable orbit 300 km above Earth

Let R= 25 c= 28 and t= 100 Find v

v= -00098l + c In R (Use the formula) v= -00098(100) + 28 In 25 (Substitute) v= -00098 + 28(3219) v= 80

Spacecraft can attain a stable orbit

Jeopardy

Exponential Vocabl Transfonnations A(t)=P(e)rt Exponential pH scale Properties of Solving Sollling Simplifying Growth

and Decay Vocabl Vocabl

Form to Logarithmic

Form

(pH= -log[H1 Logarithms Exponential Equations

of Different and

Common Bases

Logarithmic and Natural Logarithmic Equations

Natural Logarithmic Expressions

$200 Determine whether it is growth or

decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7

increase t= 10 years

Find A(t)

103=1000 What is the pH of a solution

with aW concentration of 15 x 10-3

M

What property is

used 101432-10942

How do you use your calculator doing this problem log8127

Use your calculator to evaluate e3

Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase

625= 54 What is W concentration of a solution

What property is used

1094- log16

How do you use your calculator

What is the value of log832

4e2x+2=16

decay Formula t= 35 years with a pH of doing this y=025(2)X Find A(t) 461 problem

log536

$600 Determine whether it is growth or decay and state the

Define Logarithm

y=-13(3X) Suppose you won a contest at the start of 5th grade that

deposited $3000 in an

82= 64 What is the pH of a solution with anW

concentration of2x10-2 M

What are the properties used log x29

Use the change of

base formula to evaluate

the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5

annual interest compounded continuously How much will you hallein the account

expression log3 33

when you enter high

school 4 years later

$800 Suppose you invest $500 in

a savings account that pays 35

Define Natural

Logarithmic Function

y=3(2X) A student wants to save

$8000 for college in five years How

49= 72 What is the W concentration of a solution with a pH of

What are the properties used log

4xfy

How do you set up

IOg8127 using a common


87

21n15-ln75

annual interest How

much will be in the account

after 5 years

much should be put into an account that pays 52

annual interest compounded

continuously

630 base

$1000 Suppose you invest $500 in a savings

account that pays 35

annual interest

When will the account contain at

least $6507

Define Exponential Equation

y=5(025)+5 How long would it take to

double your principal in an account that pays 65


10-2= 001 What is the pOH ofa

solution with anW

concentration of 37 X 10-7

M (Hint Find the

pH and subtract from

14 to find pOH)

What are the

properties used 610g2

x+ 510g2 y

What is the answer for

log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo

ACt) os Pe)(t Anampwect A(t) -s 200 (e)COUO)

A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)

SOOO pCe)o1~ eOlltO eO2foshy

P lt0 logtL

Ellt onQtytcu forn to LO acdhYC Form AnampNer~ t 100 0 3 l G G otI 000J 0 - 2 ~ 0 0 I

0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y

~WOO) Sl - ul1 ___ logs loY $ 2

tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)

P -- SCo e An~Ne-y-~ ~ H s - 0 9) 5 )( 0 -3 M1 O-~ 2S2

l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS

Omiddotmiddot4lJII ~[H+J [- - 1~ 2 S 4 K 0 ~ S M

P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]

0 -lo~G bull [H ~J - -II J s 0 gtlt 0 -1 1

~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e

2n lt0 - Y)liS nJcol - nl5

Y CO l 5

(6

~OOll Jn2tn~L2 e C(l 0 X - 2)

faA -- e 2shy

l b t _ Q~ 1 - ~__

Without graphing determine whether the function represents exponential growth or decay Then find the y~intercept

f(x)= 2(O65X) -exponential decay - Y-int (02)

y=3(4X)

- exponential growth - (011)

Example 3 Modeling Exponential Growth Suppose you invest $500 in a savings account that pays 35 annual interest How much will be in the account after five years

a=500 b=(1 +0035)= 1035

y=500(1035)t _ y=500(1035)5 = $59384

Example 5 Writing an Exponential Function The table shows the world population of Iberian lynx in 2003 and 2004 If the trend continues and the population is decreasing exponentially how many Iberian lynx will there be in 2020

Year Population

2003 150

2004 120

y=abX=a( 1 +r)X

1) Define the variables x= number of years since 2003 y=population of lynx

2) Solve for a and b Use y=abx

150=abo - a= 150 120= ab1

_ 120=150b - b=08

3) Find the value of r 1+r=b _1+ r= 08 - r= -02






Concept Summary




Definitions




x y=3x y=-13(3X )

-2 19 -127

-1 113 -19

0 1 -13

1 3 -1

2 9 -3





x y=2x x y=2x

-2 14 1 2

-1 12 2 4

0 1 3 8





0 203 135

5 177 109

10 153 85

15 137 69

20 121 53

25 111 43

30 104 36


y=1345(0956Y + 68




bull Recall x=bY

logbx= y



1 100=102

log10100=2

2 81=34

log381=4

3) 36=62

log636=2


8Y=32

23Y=25

3y=5 y=53

2) log5125

5Y=125 5Y=53

y=3

3 log432

4Y=32 22Y=25

2y=5 y=52
















2) y=510g2x


3)y=log2(x+4)




m=bX

mn=bx bull bY

mn=bx+y

109bmn=109bm+109bn







2) 2 10946-10949


109436-10949 (simplify 62to 36)


10944 (simplify)


3) 109432-10942


109416 (divide)


2 (simplify)







2) log39x5





log328 16

2) log418

log184 2085



Substance A 2-[H+e]

Substance B [H+e]

A pH=-log[2([H+e])]

B pH=-log[H+e]



pH of B is bigger


1) 273X= 81 333X=34


2) 53X=1125 53X= 5-3 (53= 125 5-3= 11125) 3x=-3 x= -1










Graphically y= 47X





Graphically

y=6x



T(n)= a(b)n T(n)=600000 a=1200000 b= 1+ -007=093

T(n)=1200000(093t 600000=1200000(093)n





1) log2x=-1










x2= 50000000000








In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__






Concept Summary




Definitions




x y=3x y=-13(3X )

-2 19 -127

-1 113 -19

0 1 -13

1 3 -1

2 9 -3





x y=2x x y=2x

-2 14 1 2

-1 12 2 4

0 1 3 8





0 203 135

5 177 109

10 153 85

15 137 69

20 121 53

25 111 43

30 104 36


y=1345(0956Y + 68




bull Recall x=bY

logbx= y



1 100=102

log10100=2

2 81=34

log381=4

3) 36=62

log636=2


8Y=32

23Y=25

3y=5 y=53

2) log5125

5Y=125 5Y=53

y=3

3 log432

4Y=32 22Y=25

2y=5 y=52
















2) y=510g2x


3)y=log2(x+4)




m=bX

mn=bx bull bY

mn=bx+y

109bmn=109bm+109bn







2) 2 10946-10949


109436-10949 (simplify 62to 36)


10944 (simplify)


3) 109432-10942


109416 (divide)


2 (simplify)







2) log39x5





log328 16

2) log418

log184 2085



Substance A 2-[H+e]

Substance B [H+e]

A pH=-log[2([H+e])]

B pH=-log[H+e]



pH of B is bigger


1) 273X= 81 333X=34


2) 53X=1125 53X= 5-3 (53= 125 5-3= 11125) 3x=-3 x= -1










Graphically y= 47X





Graphically

y=6x



T(n)= a(b)n T(n)=600000 a=1200000 b= 1+ -007=093

T(n)=1200000(093t 600000=1200000(093)n





1) log2x=-1










x2= 50000000000








In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__



Concept Summary




Definitions




x y=3x y=-13(3X )

-2 19 -127

-1 113 -19

0 1 -13

1 3 -1

2 9 -3





x y=2x x y=2x

-2 14 1 2

-1 12 2 4

0 1 3 8





0 203 135

5 177 109

10 153 85

15 137 69

20 121 53

25 111 43

30 104 36


y=1345(0956Y + 68




bull Recall x=bY

logbx= y



1 100=102

log10100=2

2 81=34

log381=4

3) 36=62

log636=2


8Y=32

23Y=25

3y=5 y=53

2) log5125

5Y=125 5Y=53

y=3

3 log432

4Y=32 22Y=25

2y=5 y=52
















2) y=510g2x


3)y=log2(x+4)




m=bX

mn=bx bull bY

mn=bx+y

109bmn=109bm+109bn







2) 2 10946-10949


109436-10949 (simplify 62to 36)


10944 (simplify)


3) 109432-10942


109416 (divide)


2 (simplify)







2) log39x5





log328 16

2) log418

log184 2085



Substance A 2-[H+e]

Substance B [H+e]

A pH=-log[2([H+e])]

B pH=-log[H+e]



pH of B is bigger


1) 273X= 81 333X=34


2) 53X=1125 53X= 5-3 (53= 125 5-3= 11125) 3x=-3 x= -1










Graphically y= 47X





Graphically

y=6x



T(n)= a(b)n T(n)=600000 a=1200000 b= 1+ -007=093

T(n)=1200000(093t 600000=1200000(093)n





1) log2x=-1










x2= 50000000000








In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__




x y=2x x y=2x

-2 14 1 2

-1 12 2 4

0 1 3 8





0 203 135

5 177 109

10 153 85

15 137 69

20 121 53

25 111 43

30 104 36


y=1345(0956Y + 68




bull Recall x=bY

logbx= y



1 100=102

log10100=2

2 81=34

log381=4

3) 36=62

log636=2


8Y=32

23Y=25

3y=5 y=53

2) log5125

5Y=125 5Y=53

y=3

3 log432

4Y=32 22Y=25

2y=5 y=52
















2) y=510g2x


3)y=log2(x+4)




m=bX

mn=bx bull bY

mn=bx+y

109bmn=109bm+109bn







2) 2 10946-10949


109436-10949 (simplify 62to 36)


10944 (simplify)


3) 109432-10942


109416 (divide)


2 (simplify)







2) log39x5





log328 16

2) log418

log184 2085



Substance A 2-[H+e]

Substance B [H+e]

A pH=-log[2([H+e])]

B pH=-log[H+e]



pH of B is bigger


1) 273X= 81 333X=34


2) 53X=1125 53X= 5-3 (53= 125 5-3= 11125) 3x=-3 x= -1










Graphically y= 47X





Graphically

y=6x



T(n)= a(b)n T(n)=600000 a=1200000 b= 1+ -007=093

T(n)=1200000(093t 600000=1200000(093)n





1) log2x=-1










x2= 50000000000








In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__


bull Recall x=bY

logbx= y



1 100=102

log10100=2

2 81=34

log381=4

3) 36=62

log636=2


8Y=32

23Y=25

3y=5 y=53

2) log5125

5Y=125 5Y=53

y=3

3 log432

4Y=32 22Y=25

2y=5 y=52
















2) y=510g2x


3)y=log2(x+4)




m=bX

mn=bx bull bY

mn=bx+y

109bmn=109bm+109bn







2) 2 10946-10949


109436-10949 (simplify 62to 36)


10944 (simplify)


3) 109432-10942


109416 (divide)


2 (simplify)







2) log39x5





log328 16

2) log418

log184 2085



Substance A 2-[H+e]

Substance B [H+e]

A pH=-log[2([H+e])]

B pH=-log[H+e]



pH of B is bigger


1) 273X= 81 333X=34


2) 53X=1125 53X= 5-3 (53= 125 5-3= 11125) 3x=-3 x= -1










Graphically y= 47X





Graphically

y=6x



T(n)= a(b)n T(n)=600000 a=1200000 b= 1+ -007=093

T(n)=1200000(093t 600000=1200000(093)n





1) log2x=-1










x2= 50000000000








In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__














2) y=510g2x


3)y=log2(x+4)




m=bX

mn=bx bull bY

mn=bx+y

109bmn=109bm+109bn







2) 2 10946-10949


109436-10949 (simplify 62to 36)


10944 (simplify)


3) 109432-10942


109416 (divide)


2 (simplify)







2) log39x5





log328 16

2) log418

log184 2085



Substance A 2-[H+e]

Substance B [H+e]

A pH=-log[2([H+e])]

B pH=-log[H+e]



pH of B is bigger


1) 273X= 81 333X=34


2) 53X=1125 53X= 5-3 (53= 125 5-3= 11125) 3x=-3 x= -1










Graphically y= 47X





Graphically

y=6x



T(n)= a(b)n T(n)=600000 a=1200000 b= 1+ -007=093

T(n)=1200000(093t 600000=1200000(093)n





1) log2x=-1










x2= 50000000000








In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__



m=bX

mn=bx bull bY

mn=bx+y

109bmn=109bm+109bn







2) 2 10946-10949


109436-10949 (simplify 62to 36)


10944 (simplify)


3) 109432-10942


109416 (divide)


2 (simplify)







2) log39x5





log328 16

2) log418

log184 2085



Substance A 2-[H+e]

Substance B [H+e]

A pH=-log[2([H+e])]

B pH=-log[H+e]



pH of B is bigger


1) 273X= 81 333X=34


2) 53X=1125 53X= 5-3 (53= 125 5-3= 11125) 3x=-3 x= -1










Graphically y= 47X





Graphically

y=6x



T(n)= a(b)n T(n)=600000 a=1200000 b= 1+ -007=093

T(n)=1200000(093t 600000=1200000(093)n





1) log2x=-1










x2= 50000000000








In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__






2) log39x5





log328 16

2) log418

log184 2085



Substance A 2-[H+e]

Substance B [H+e]

A pH=-log[2([H+e])]

B pH=-log[H+e]



pH of B is bigger


1) 273X= 81 333X=34


2) 53X=1125 53X= 5-3 (53= 125 5-3= 11125) 3x=-3 x= -1










Graphically y= 47X





Graphically

y=6x



T(n)= a(b)n T(n)=600000 a=1200000 b= 1+ -007=093

T(n)=1200000(093t 600000=1200000(093)n





1) log2x=-1










x2= 50000000000








In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__


1) 273X= 81 333X=34


2) 53X=1125 53X= 5-3 (53= 125 5-3= 11125) 3x=-3 x= -1










Graphically y= 47X





Graphically

y=6x



T(n)= a(b)n T(n)=600000 a=1200000 b= 1+ -007=093

T(n)=1200000(093t 600000=1200000(093)n





1) log2x=-1










x2= 50000000000








In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__




Graphically

y=6x



T(n)= a(b)n T(n)=600000 a=1200000 b= 1+ -007=093

T(n)=1200000(093t 600000=1200000(093)n





1) log2x=-1










x2= 50000000000








In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__







x2= 50000000000








In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__







In125


In 18



x= e2

Check Answer

In e2 = 2

2=2


In 6x= 2



Check Answer

In 2(e26) + In 3= 2

1m 6e26 = 2 2=2



2) e2x= 12


x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__

x= (In 12)2





Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__

Jeopardy



Form to Logarithmic

Form


of Different and

Common Bases




decay y=12(O95)X

Define Exponential

growth

y=2(X-4) p= $131 r-7


Find A(t)



M

What property is

used 101432-10942



Inx=2


Define Change of

Base

y=3(x+5) p= $5000 r- 10

increase



1094- log16



4e2x+2=16


log536


Define Logarithm






Use the change of


the


In(x-3)2=4

y-intercept y=3(4X)

account that pays 5


expression log3 33

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__

when you enter high




Define Natural






4xfy

How do you set up



87

21n15-ln75

annual interest How


after 5 years



continuously

630 base



annual interest


least $6507






solution with anW


M (Hint Find the


14 to find pOH)

What are the


x+ 510g2 y


log8127


In2x+ln3=2

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__

~ZCO)

t400)

cI ~ i) (gOO)

$poundCraquo


A(b) os 402 gt 1 5

A(t) - 500G(Q)o ()(3S)

ACt) S I (c5Sll 2(0

A( ) 005(4) t 3000(2)r

ACt) 3lc (04 2l

8000 PCe)O052(S)


P lt0 logtL


0 9 LO 00 gt ~ og2 0 61 ~ ~ gt

5 4ampLOO) (o2~ ~

09sQ2~ Y


tSOC) yq s 12

log 4q - 2

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__

$ 200)

ltamp - 00)

ltamp(cpoundO

~~)

__ 1 000)


l loA ~ - 09 CH ~J - l -

-l1lQJ 09[-1 tJS


P l- S - tog [2 x 0 - 2 M1 ~H ~ 10

lt0 30 ~ _~[l-4 J - l - l

-lt0 30 09 [H 1 ]


~ - s - 0 9[3 1 x 0 -1 -1 )- ~ (0 3 PO - s L 0 - (0 I 4 3 ~ t S1

shy

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__

Ye$YCO Y e2

l -1 2 ~lo 2

)( =111 e2t =-3S

2gtlt 13gt X - 103 5

2shy

S (000( (y ~hl74 ( ~A

A~6~middot eJ l

~l- e


Y CO l 5

(6


faA -- e 2shy

l b t _ Q~ 1 - ~__

Date post:	01-Aug-2020
Category:	Documents
Upload:	others
View:	4 times
Download:	0 times

Unit 8: ExPonential! Logs · • For exponential growth y=abx, with b>1 the value b is the growth...

Documents