R—05/28/09—HW #73: Pg 477:47,49,50; Pg 490:17,49-61odd; Pg 496:31-55 eoo; Pg 505:25-59 odd
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Transcript of R—05/28/09—HW #73: Pg 477:47,49,50; Pg 490:17,49-61odd; Pg 496:31-55 eoo; Pg 505:25-59 odd
R—05/28/09—HW #73: Pg 477:47,49,50; Pg 490:17,49-61odd; Pg 496:31-55 eoo; Pg 505:25-59 odd50) V=22000(.875)^t; 14,738
F—05/29/09—HW #74: Pg 469: 19-24; Pg 477: 19-24; Pg 496: 33-57 eoo; Pg 505: 26-58 even20) E 22) A 24) D
20) E 22) B 24) A
26) -1\5 28) no sol 30) ln(1\6) 32) (ln 3)\534) ln 5\3 36) -.661 38) (ln5.5)\4 40) ln13.542) log255 44) 1\2 46) e^(15\8) 48) e^(5\2)50) 15 52) .729 54) 256 56) no sol58) no sol
Chapter 8 Review
xy 2x
y
2
1
An exponential function involves y = bx where the base b is positive and not equal to 1.
An asymptote is a line the graph approaches.
Usually represented by a dotted line
An exponential growth function is when b > 1
An exponential decay function is when 0 < b < 1
Tell me if the following are growth or decay functions.
xy 2x
y
2
12
xy 24x
y
3
5
6
1
Decay Growth Growth
x
y
2
14 Decay
One to Review
1) Find key points
2) Add ‘h’ to the x-value
3) Multiply ‘a’ to the y-value
4) Add ‘k’ to the y-value
242 1 xy )4,1()1,0(4
1,1
Other info to find:
y – intercept:
Asymptote:
Domain: All real numbers
Range:
End behavior: x ∞; f(x) _____
x -∞; f(x) _____
h = 1)4,2()1,1(
4
1,0
a = 2 )8,2()2,1(2
1,0
k = -2)6,2()0,1(
2
3,0
2
3242 1
2y
),2(;2 y2
A town of 5000 grows at a rate of 10% per year. How many people are in town after 10
years?
y = a(1 + r)t
y final value
a starting value
r % increase, in decimal form
(1 + r) Growth Factor
t Time
Find the info
Write equation
Answer question
5000
.1
1.1
10
5000
.1
1.1
10
y = (1 + )Calculator Plug – In
5000 * (1.1)^10
12968 people
(answers must make sense)
A car that costs 25,000 depreciates at a rate of 5% per year.
A) What is the value of the car after 7 years?
B) When is the car worth $20,000?
Calculator – Windows and Intersect
7)^95(.25000
)05.1(25000 7y
43.17458$
y = a(1 – r)t
a
r
Decay factor
t
25000
.05
.95
7
20000
)95(.25000
y
y x
You can treat it like a number.
23 ee 23e
5e
xe
e5
3
xe 53
43 )( xe
43 xe
xe12
13 )2( xe)1(31)2( xe
22
33 xx ee
3 6xe3 32 )( xe
xe2
Let b and y be positive numbers, and b = 1. The LOGARITHM of y with base b is logby and is defined
as
logby = x if and only if bx = y
You read this log base b of y equals x.
log 125 35
log
22
1
4 125 35
22
1
4
Rewrite in Exponential Form
01log5 14log4 29log3
1 2
532log4
4log2isWhat
?42 equalspowerwhatto
2
8log2
1isWhat
?82
1equalspowerwhatto
3
100log10isWhat
?10010 equalspowerwhatto
2
10
10
loglog
logarithmcommon
10
log,10
log
means
acalledisBase
x
baseisbuttonthe
calculatoryourOn
e
e
means
acallediseBase
xe
baseisbuttonthe
calculatoryourOn
logln
logarithmnatural
log,
ln
54.4ln
ln
20log
e
3010.1
1
5129.1
Log Properties: g(x) = logbx is the inverse of f(x) = bx
That means g(f(x)) = logbbx = x, and f(g(x)) = xb xb log
Try to make things match up with the base, and it’ll work out ok.
x10log10x3log3
xe 2ln x27log3
27
1log3isWhat
e
Calc
log
Finding inverses
A) Simplify first
1) Switch x and y
2) Change forms
3) Simplify
)3ln( xyy x
)3ln( e
3
yex 3
yx 3log yx )4log(
Finding inverses
A) Simplify first
1) Switch x and y
2) Change forms
3) Simplify
6021.4log 0792.112log
48log
124log
12log4log
6813.1
0792.16021.
nun
vuv
u
vuuv
rules
bu
b
bbb
bbb
loglog
logloglog
logloglog
!log
3log
4
12log
4log12log
4771.
6021.0792.1
3
1log
12
4log
12log4log
4771.
0792.16021.
16log
24log
4log2
2042.1
)6021(.2
144log
212log
12log2
1584.2
)0792.1(2
!log ruleswithSolve
24log x
2log4log x
xlog24log
!expandorCondense
2log10log
2
10log
onenttoroot
changeClue
exp
,5log
3
10log
15log3log2log x )log210(log6log2
1y
2
1
5log
5log2
13log10log
5log
15log3loglog 2 x15log9loglog x
15log9log x
15
9log
x
5
3log
x
)log10(log6log 22
1
y
)10(log6log 22
1
y
2
2
1
10
6log
y
3log1
• General Solving Rules• Methods of solving
– Make bases of exponents the same• Notice when both sides have x as an exponent, and it looks like you
can make bases the same.
– Log both sides• Generally if you have a variable exponent on only one side.• Make the base of your log the same as the base of the exponent
– Make terms inside logs equal• Both sides of the equation have logs with same bases• May involve condensing log expressions
– Exponentiating both sides• One side has a log, one side doesn’t• May involve condensing log expressions
– NEED TO DOUBLE CHECK!!!!
x225
!exp
,
samethebemustonentsso
sametheisBase
x5
41 22 x
41x
3x
124 xx
12 xx
1x
12 2)2( xx
61 24 x
622 x
42 x
6)1(2 2)2( x
2x
83255
1 xx
235 162 xx
7
16
7
8
64 5 x
636 x
922 x
72 x
1427 3 x
8074.2
2925.6
2263.1
5850.1
7230.1
)4(log)2(log 33 xx xx 42 x32
x3
2
)10ln()2ln( xx
102 xx
10x
)2(log)32(log 66 xx
232 xx
1x
)5(log)63(log 33 xx xx 563 x26 x 3
CAN’T HAVE NEGATIVE INSIDE OF LOGS! EXTRANEOUS SOLUTION.
MUST ALWAYS DOUBLE CHECK!!!!!
Solution
NO
2log3 x
2log 33 3 x
23x
9x
9)2(log3 4 x
Convert 3)2(log 44 4 x
342 x66x
3)2(log4 xMUST ALWAYS DOUBLE CHECK!!!!!
Exponentiating both sides
2)1(log)2(log 22 xx
2))1)(2((log2 xx
062 xx
2,3
0)2)(3(
xx
xx
2)ln()1ln( xx
2)ln( 2 xx
22 exx
)1(2
))(1(4)1()1( 22 ex
2)2(log 22 xx
2)2(log 222
2 xx
422 xx
2)ln( 2
ee xx
022 exxwork
tdoesnanswerOne
eApproximat
'
formula
quadraticntHi ,
MUST ALWAYS DOUBLE CHECK!!!!!
• R—05/28/09—HW #73: Pg 477:47,49,50; Pg 490:17,49-61odd; Pg 496:31-55 eoo; Pg 505:25-59 odd
• F—05/29/09—HW #74: Pg 469: 19-24; Pg 477: 19-24; Pg 496: 33-57 eoo; Pg 505: 26-58 even
•