we can unite bases!
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We can unite bases!. Now bases are same!. We can unite bases!. Now bases are same!. Check (Remember: Back to Original) . We can unite bases!. Now bases are same!. 8-4 Solving Logarithmic Equations and Inequalities . Attention Inequality log Domain first. - PowerPoint PPT PresentationTRANSCRIPT
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3 243x
5x
3log 243Solve xRe :logb
membera c
ca b53 3x
We can unite bases! Now bases are same!
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1 148
xSolve We can unite bases!2( 1) 12 8x
2( 1) 3( 1)2 2x 2 2 32 2x Now bases are same!
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2 2 3x 2 2 3x 2 3 2x 2 5x
52
x
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Check (Remember: Back to Original) 52
1 148
x
215 148
1 18 8
true
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21927
x
xSolve
We can unite bases!2 23 27x x 2 3( 2)3 3x x 2 3 63 3x x Now bases are same!2 3 6x x
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2 3 6x x 5 65 5x
65
x
Check in original219
27
x
x
66 55
21927
13.9666 13.9666
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8-4 Solving Logarithmic Equations and Inequalities
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2 2log 3 log 2 1Solve x x
Attention Inequality log Domain first. 3 0Domain x 3x
2 1 0x 2 12 2x
0.5x
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2 2log 3 log 2 1x x
3 2 1x x 2 1 3x x
2x
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21 1x
2x
Reverse the direction when dividing by “minus”:{ 0.5 2}solution x
3x 0.5x From domain before
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Check 1 (Remember: Back to Original) 2 2log 3 log 2 1x x
2 2log 3 lo1 (g )2 11
2 2log 4 log 3
2 1.5850 true
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2 2log 2 log 6 3Solve x x
Attention Inequality log Domain first. 2 0Domain x 2x
6 3 0x 3 63 3x
2x
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2 2log 2 log 6 3x x 2 6 3x x 3 6 2x x 4 44 4x
1x 2x 2x
From domain:
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:{1 2}solution x Check 1.5 (Remember: Back to Original) 2 2log 2 log 6 3x x 2 2log 2 log 61.5 (1.5)3
2 2log 3.5 log 1.51.8074 0.5850 true
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3 3log 3 4 log 2Solve x x
Attention Inequality log Domain first.3 4 0Domain x 43
x
2 0x 2x
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3 3log 4 3 log 2x x 4 3 2x x 4 2 3x x
3 53 3x
53
x
43
x
2x
From domain::{ 2}solution x
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15
log 125Solve x Re :logb
membera c
ca b
11255
x
We can unite bases!
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35 5 x Now bases are same!3 x
3x
11255
x
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11log 7 1Solve x Attention Inequality log Domain first. 7 0Domain x
7x 11log 7 1x
Re :logb
membera c
ca b17 11x
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7 11x 11 7x 4x
7x From domain
{ 4 7}solution x
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Check 0 (Remember: Back to Original) 11log 7 1x
11 0log 7 1
11log 7 1
0.8115 1 true
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8-5 Properties of Logarithms
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3 3log 2 log 2Solve x x
Re :
log log logb b b
membermm nn
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3 3log 2 log 2x x
3
2log 2xx
22 3x
x
2 9xx
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91
2xx
2 9x x 2 9x x 2 8x
14
x
Do Cross Multiply
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: ( )14
Check replace in original
3 3log 21 1o42
4l g
3 3
9 1log log 24 4
2 2 true
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2 2log 4 5 logSolve x x
2 2log 4 log 5x x
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Re :log log logb b b
memberm n m n
2 2log 4 log 5x x
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2log 4 5x x
54 2x x
2 2log 4 log 5x x
2 4 32x x 2 4 32 0x x
Re :logb
membera c
ca b
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2 4 32 0x x 4 8 0x x
Use MODE 5 3 a = 1, b= -4, c= -32
4 04
xx
8 08
xx
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Check -4 (Remember: Back to Original) 2 2log 4 5 logx x 2 2log 4 5 log4 4
Undefined, so ignore -4
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Check 8 2 2log 4 5 logx x 2 2log 4 g8 5 8lo
2 5 3 true 2 2log 4 5 log 8
only solution is 8
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2 2log 2 3 2logSolve x x
2 2log 2 3 2logx x 2 2
2log 2 3 logx x 22 3x x
2 2 3 0x x
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2 2 3 0x x Use MODE 5 3 a = 1, b= -2, c= -3
3 1 0x x 3 03
xx
1 01
xx
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Check 3 (Remember: Back to Original) 2 2log 2 3 2logx x
2 2(3) 3log 2 3 2log
2 2log 9 2log 33.1699 = 3.1699
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Check -1 (Remember: Back to Original) 2 2log 2 3 2logx x
2 2log 2 3 2log( 1) ( 1)
Undefined, so ignore -1only solution is 3
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2
2 2log 9 log 4 6Solve m
Re :log log logb b b
memberm n m n
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2
2 2log 9 log 4 6m 2
2log 4 9 6m
Re :logb
membera c
ca b
2 64 9 2m 24 36 64m
24 100m
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24 100m 2 25m 2 25m
5m
Square root both sides
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Check -5 (Remember: Back to Original) 2
2 2log 9 log 4 6m
2
2 2log 9 log 45 6
2 2log 16 log 4 6 6 6 true
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Check 5 (Remember: Back to Original) 2
2 2log 9 log 4 6m 2 2
2log 9 log 4 65
2 2log 16 log 4 6 6 6 trueThe solutions are 5 and -5
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2 2 2 2log 3 log log 4 log 4x x Solve. Check your solution.
2 2 2 2log 3 log log 4 log 4x x
Re :
log log logb b b
membermm nn
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2 2 2 2log 3 log log 4 log 4x x
2 2
3 4log log4x x
3 44x x
3( 4) 4x x
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3( 4) 4x x
3 12 4x x
12 4 3x x
12x
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Check 12 (Remember: Back to Original) 2 2 2 2log 3 log log 4 log 4x x
2 2 2 2log 3 log log 4 log 1 412 2
2 2 2 212log 3 log log 4 log 16 2 2 true
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2
4 4 4log 4 log 2 log 1x x Solve. Check your solution.Re :
log log logb b b
membermm nn
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2
4 4 4log 4 log 2 log 1x x 2
4 4
4log log 12
xx
2 4 12 1
xx
2 4 2x x
Do Cross Multiply
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2 4 2 0x x 2 6 0x x Use MODE 5 3
a = 1, b= -1, c= -6 3 2 0x x
3 03
xx
2 02
xx
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Check 3 (Remember: Back to Original) 2
4 4 4log 4 log 2 log 1x x
2
4 4 4log 4 log og3 3 2 l 1
4 4 4log 5 log 5 log 1
0 0 true
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Check -2 (Remember: Back to Original) 2
4 4 4log 4 log 2 log 1x x
4 4 4
2log 4 log( 2) ( 2) 2 log 1
4 4 4log 0 log 0 log 1 Undefined, so ignore -2only solution is 3
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log 12 ?a . 2log 2 log 3b bA . log 5 2log 2a aB
. log 14 log 2a aC
. log 3 2log 2a aD
log 12 ( )b
log 20 ( )a
log 7 ( )a
log 12 ( )a
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2 2 2
1 1log log 16 log 254 2
Solve m
1 14 2
2 2 2log log 16 log 25m Raise the powers
1 14 2
2 2log log 16 25m
2 2log log 10m
10m
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4 4 4 4
1log 0.25 3log log 64 5log 23
Solve x
Raise the powers13 53
4 4 4 4log 0.25 log log 64 log 2x 1
3 534 4log 0.25 log 64 2x
3
4 4log 0.25 log 128x 3
0.250.25 1
25280.
x
3 512x 33 3 512x
8x
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9log 5 log 29 bEvaluate and b9log 595
log 2bb2
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5 512log 3 log 2735Evaluate
12 3
5 5log 3 log 275
2
5 13
3log
2755log 35 3
Raise the powers first!
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1 1log 4 log 272 3b bEvaluate b
1132
5 5log 4 log 27b
1132log 4 27bb
log 6bb6
Raise the powers first!
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3 3 3 3log 5 log 10 log 4 log 2Show that
3 3 3 3 3log 5 log 10 log 4 5log log 410
3
5log 410
3log 2
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8-6 Common Logarithms
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Express log9 22 in terms of common logarithms. Then approximate its value to four decimal places.9
log22log 22log9
1.4068
Common logarithm change to base 10
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Express log5 14 in terms of common logarithms. Then approximate its value to four decimal places.5
log14log 14log5
0.6099
Common logarithm change to base 10
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25 21xSolveRound to four decimal places
We can’t unite bases!So, “log” both sides!2log 5 log21x
2 log 5 log21x
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2 log 5 log21x Divide by 2log5 !!2log52 log5 log21
2log5x
0.9458x
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34 10xSolveRound to four decimal places
We can’t unite bases!So, “log” both sides!3log 4 log10x
3 log 4 log10x
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3 log 4 log10x Divide by 3log4 !!3log43 log4 log10
3log4x
0.5537x
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36 5xSolve We can’t unite bases!So, “log” both sides!A. 0.2375
B. 1.1132C. 3.3398D. 43.2563
Do the calculations!
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3 3log 2 0.6309 log 12Use toapproximate
3 3log 12 log 2 2 3
3 3 3log 2 log 2 log 3 0.6309 0.6309 1 2.2618
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3 3
3log 2 0.6309 log2
Use toapproximate
3 3 3
3log log 3 log 22
1 0.6309 0.3691
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5log 11 1.4899Use and5 5log 2 0.4307 log 44to find
5 5log 44 log 2 2 11
5 5 5log 2 log 2 log 11
0.4307 0.4307 1.4899 2.3513
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5log 3 0.6826Use and5 5log 2 0.4307 log 54to find
5 5log 54 log 2 3 3 3
5 5 5 5log 2 log 3 log 3 log 3
0.4307 0.6826 0.6826 0.6826
2.4785
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4log 3 0.7925Use and
4 4
9log 7 1.4037 log7
to find
4 4 4
9log log 9 log 77
4 4log 3 3 log 7 4 4 4log 3 log 3 log 7 0.7925 0.7925 1.4037 0.1823
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4log 3 0.7925Use and
4 4
7log 7 1.4037 log12
to find
4 4 4
7log log 7 log 1212
4 4log 7 log 3 4 4 4 4log 7 log 3 log 4
1.4037 0.7925 1 0.3888
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Solve. Round to four decimal places.2 3 34 9x x We can’t unite bases! So give “log”2 3 3log4 log9x x 2 3 log4 3 log9x x
2 log4 3log4 log9 3log9x x
2 log4 log9 3log9 3log4x x
2log4 log9 3log9 3log4x
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2log4 log9 3log9 3log4x
2log4 log92log4 log9 3log9 3lo
2g4
log4 log9x
3log9 3log2log4 log9
4x
4.2283x
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1Pr logloga
b
ove ba
lo. . gal h s b
.1log
.b a
r h s
We change L.H.S to base “b”1log
loga
b
ba
loglog
b
b
ba
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Challenge Evaluate 3 3
2 5log 5 log 2
2 53log 5 3log 2
2 53 3 log 5 log 2
2 5log 5 log 29 199
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8-7 Natural Logarithms
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Remember! ln xe x ln xe xln 210Evaluate eln 210 e
10 2 8
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5ln6 8Solve x First isolate the “ln” then give it base “e”5ln6 85 5
x
8ln65
x 8
ln6 5xe e
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856
6 6x e
0.8255x
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ln(6 3) 3 10Solve x First isolate the “ln” then give it base “e”ln(6 3) 7x
ln(6 3) 7xe e 76 3x e 76 3x e 7 36
ex 183.2722x
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53 1 10xSolve e 53 10 1xe 53 93 3
xe
First isolate the “e” then “ln” both sides5 3xe 5n l 3l nxe 5 ln3x
ln35
x
0.2187x
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24 5 1xSolve e 24 5 1xe
24 64 4
xe
First isolate the “e” then “ln” both sides2 3
2xe
2n n 32
l lxe
32 ln2
x
3ln22
x
0.2187x
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2ln 5xSolve e 2ln 5xe 2 5x 5 2x 3x
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Write each exponential in logarithmic form2xe “ln” both sidesln ln 2xe
ln 2x
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Write each exponential in logarithmic form0.35x e “ln” both sides0.35ln lnx e
ln 0.35x
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Write each logarithm in exponential formln 0.6742x “e” both sidesln 0.6742xe e0.6742x e
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Write each logarithm in exponential formln 22 x “e” both sidesln 22 xe e
22 xe
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Write each expression as a single logarithm4ln9 ln 274ln9 ln 2749ln27
ln 2435ln3
5ln3
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Write each expression as a single logarithm17ln 5ln 22
7
51ln ln 22
7
51ln 22
7 5ln 2 2
2ln 2
2ln 2
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Challenge Evaluate ln5
3 3log 24 log 8e ln5
3 3log 24 log 8 e
ln5
3
24log8
e
ln5
3log 3 e
1 5 6
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Challenge Solve 5 5log 2 log 3 45 lnx x xe 52log35 4x
x x 2 43x x
x
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2 43 1x x
x
2 3 4x x x 22 12x x x
2 12 0x x 3 4 0x x
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3 4 0x x 3 4x x Check -3 5 5log 2 log 3 45 lnx x xe
5 5log 2 log 3( 3) 3 435 ln e undefinedCheck 4 5 5log 2 log 4 3(4 4) 45 ln e 8 8 true
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7-1 Operations on Functions
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3 1 3 14 64x xSolve We can unite bases!3 1 3(3 1)4 4x x
3 1 9 34 4x x Now bases are same!3 1 9 3x x
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3 9 3 1x x 6 4x 6 46 6x
23
x
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Compound Interest You deposited $700 into an account that pays an interest rate of 4.3% compounded monthly.How much will be in the account after 7 years?12n
7t 700P
1ntrA P
n
0.043r
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1ntrA P
n
12 70.043700 112
A
$945.34A
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Compound Interest You deposited $1000 into an account that pays an interest rate of 5% compounded quarterly.a) How much will be in the account after 5 years?4n
5t 1000P
1ntrA P
n
0.05r
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1ntrA P
n
4 50.051000 14
A
$1282A
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Compound Interest You deposited $1000 into an account that pays an annual rate of 5% compounded quarterly.b) How long it take until you have a $1500 in your account?1
ntrA Pn
1500A1000P
?t
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1ntrA P
n
40.051500 1000 14
t
41500 1000 1.0125 t Divide both sides by 1000
41.5 1.0125 t
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41.5 1.0125 t “log” both sides now4log1.5 log1.0125 t
log1.5 4 log1.0125t
log1.5 4 log1.0124log1.0125 4log1. 25
501
t
8.16t yrs
Divide both sides by 4log1.0125
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1( ) 2 3xGraph f x
X Y -2 3.125 -1 3.25 0 3.5 1 4 2 5
Use MODE 7{ }Domain All real numbers
{ 3}Range y
: 3Asymptote y
int : 0, 3.5y ercept
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1( ) 2 3xGraph f x
X Y -2 3.125 -1 3.25 0 3.5 1 4 2 5
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1( ) 22
x
Graph f x
X Y -2 8 -1 4 0 2 1 1 2 0.5
Use MODE 7{ }Domain All real numbers
{ 0}Range y
: 0Asymptote y
int : 0, 2y ercept
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X Y -2 8 -1 4 0 2 1 1 2 0.5
1( ) 22
x
Graph f x
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{ 0}Domain x { }Range All real numbers
: 0x
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2( ) logGraph f x xPoints:(1, 0)(2, 1)1 , 12
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{ 0}Domain x { .}Range All real no
: 0Asymptote x
2( ) logGraph f x x
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{ 0}Domain x { .}Range All real no
: 0Asymptote x
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3( ) log 2Graph f x x Shift 2units upPoints:(1, 0)(3, 1)1 , 13
After shift:(1, 2)(3, 3)1 , 13
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3( ) log 2Graph f x x
{ 0}Domain x
{ }Range All real numbers
: 0Asymptote x
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X=2X=-3
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2( ) log ( 1)Graph f x x Shift 1unit rightPoints:(1, 0)(2, 1)1 , 12
After shift:(2, 0)(3, 1) 1.5, 1
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2( ) log ( 1)Graph f x x
:{ 1}Domain x
{ .}Range All real no
: 1Asymptote x
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2( ) log ( 3) 1Graph f x x
Shift 3units left and 1 unit upPoints:(1, 0)(2, 1)1 , 12
After shift:(-2, 1)(-1, 2) 2.5,0
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2( ) log ( 3) 1Graph f x x
:{ 3}Domain x
{ .}Range All real no
: 3Asymp x
X=-3
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Write an exponential function whose graph passes through the points (0, 15) and (3, 12)xy ab015 ab 15 a
Now replace second point and also “a=15”312 15b312 1
5 55
1 1b 3
12 0.9315
b 15(0.93) xy
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Write an exponential function whose graph passes through the points (0, 256) and (4, 81) xy ab0256 ab 256 a Now replace second point and also “a=256”481 256b
481 2556 5
62 2 6
b 481 3256 4
b
32564
x
y
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Exponential growth with given rate: 1 ty a r A house was bought for $96,000 in the year 2000. The house appreciates at a rate 7%. 1) Write an exponential equation that models the price after t years. 1 ty a r 96000 1 0.07 ty
96000 1.07 ty
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2) Find the price in the year 2003. 96000 1.07 ty
396000 1.07y 117604.128y
$117,604.128price will be