Chapter 4 – Exponential and Logarithmic Functions 4.3 - Logarithmic Functions.
Logarithmic Properties & Functions
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Logarithmic Properties & Functions By: Jeffrey Bivin Lake Zurich High School [email protected] Last Updated: January 30, 2008
description
Logarithmic Properties & Functions. By: Jeffrey Bivin Lake Zurich High School [email protected]. Last Updated: January 30, 2008. Properties of Logarithms. Product Property Quotient Property Power Property Property of Equality. Jeff Bivin -- LZHS. Product Property. multiplication. - PowerPoint PPT Presentation
Transcript of Logarithmic Properties & Functions
Logarithmic Properties &
FunctionsBy: Jeffrey Bivin
Lake Zurich High School
Last Updated: January 30, 2008
Properties of Logarithms
• Product Property
• Quotient Property
• Power Property
• Property of Equality
Jeff Bivin -- LZHS
Product Property
nmnm aaa
)(log)(log)(log nmnm bbb multiplication addition
multiplication addition
Jeff Bivin -- LZHS
Product Property
)4(log)16(log)416(log 222
)2(log)2(log)22(log 22
42
242
24)2(log 62
66
Jeff Bivin -- LZHS
Quotient Property
nmn
m
aa
a
)(log)(log)(log nm bbnm
b division subtractio
n
division subtraction
Jeff Bivin -- LZHS
Quotient Property
)4(log)32(loglog 22432
2
)2(log)2(log8log 22
522
25)2(log 32
33
Jeff Bivin -- LZHS
Power Property
nmnm aa
logb(m p )
logb(mp ) = p•logb(m)
p
Jeff Bivin -- LZHS
Power Property
)2(log72log 27
2
177
77
Jeff Bivin -- LZHS
Property of Equality
CAthen
)(log)(log CAif bb
Jeff Bivin -- LZHS
)(log 235 yx
Expand
)(log)(log 25
35 yx
)(log2)(log3 55 yx
product property
power property
Jeff Bivin -- LZHS
Expand
)(log)(log 45
355 zyx
)(log)(log)(log 45
35
55 zyx
quotient property
product property
)(log4)(log3)(log5 555 zyx power property
4
35
5logz
yx
Jeff Bivin -- LZHS
)(log)(log)(log 55
25
75 zyx
)(log)(log 525
75 zyx
Expand
quotient property
product property
)(log5)(log2)(log7 555 zyx power property
52
7
5logzyx
)(log)(log)(log 55
25
75 zyx distributive
property
Jeff Bivin -- LZHS
zyx 333 log2log6log5
Condense
power property
product property
23
63
53 logloglog zyx
23
653 loglog zyx
2
65
3logz
yxquotient property
Jeff Bivin -- LZHS
410
21010 logloglog 2
1
zyx
zyx 10101021 log4log2log
Condense
group / factor
product property
4102
1010 logloglog 21
zyx
421010 loglog 2
1
zyx
42
21
10logzyxquotient
property
Power property
4210logzy
x
Jeff Bivin -- LZHS
4523 loglogloglog wzyx eeee
4253 loglogloglog wyzx eeee
wzyx eeee log4log5log2log3
Condense
re-organizegroup
4253 loglogloglog wyzx eeee
4253 loglog wyzx ee
42
53
logwyzx
e
product property
Power property
quotient property
Jeff Bivin -- LZHS
Solve for x
393 xx
122 x
6x
3log93log 33 xx
Jeff Bivin -- LZHS
Property of Equality
3log93log 33 xx
Solve for x
6xcheck
36log9)6(3log 33
36log918log 33
9log9log 33
checks!
Jeff Bivin -- LZHS
3log93log 33 xx
Solve for x
393 xx
122 x
6x
6Jeff Bivin -- LZHS
nn 6log2log7log 444
Solve for n
nn 6147
14n
nn 6log)2(7log 44
Jeff Bivin -- LZHS
Condense left side
Property of Equality
nn 6log2log7log 444
Solve for n
14ncheck
)14(6log214log7log 444
84log12log7log 444
84log)12(7log 44
84log84log 44 checks!
Jeff Bivin -- LZHS
nn 6log)2(7log 44
nn 6log2log7log 444
Solve for n
nn 6147
14n
14Jeff Bivin -- LZHS
Solve for x
31log1log 22 xx
)1)(1(23 xx
18 2 x29 xx3
3)1)(1(log2 xx
Jeff Bivin -- LZHS
Condense left side
Convert to exponential
form
Solve for x
31log1log 22 xx
3xcheck 3xcheck
313log13log 22
32log4log 22
312 33
checks!
313log13log 22
34log2log 22
fails
The argument must be positive
Jeff Bivin -- LZHS
Solve for x
3)1)(1(log2 xx
)1)(1(23 xx
18 2 x29 xx3 3
31log1log 22 xx
Jeff Bivin -- LZHS
Solve for x
33 22 xx
39 2 xx
60 2 xx
)2)(3(0 xx
23log 23 xx
23 xorx
Jeff Bivin -- LZHS
Convert to exponential
form
23log 23 xx
Solve for x
3xcheck
checks!
23)3()3(log 23
2339log3 29log3
22
2xcheck
232)2(log 23
2324log3 29log3
22
checks!
Jeff Bivin -- LZHS
23log 23 xx
Solve for x
33 22 xx
39 2 xx
60 2 xx
)2)(3(0 xx23 xorx 2,3
Jeff Bivin -- LZHS
Solve for x
)19log()5log()73( x
)19log()5log(7)5log(3 x
)5log(7)19log()5log(3 x
)5log(3
)5log(7)19log( x
943.2x
19log5log 73 x
)5log(3)5log(3
Jeff Bivin -- LZHS
24 7log5log x
Solve for x
)7log()2()5log()4( x
)5log(4
)7log(2x
605.0x
)5log(4)5log(4
Jeff Bivin -- LZHS
Solve for x
)9log()5()11log()12( xx
)9log(5)11log(1)11log(2 xx
)11log()9log(5)11log(2 xx
)11log()9log(5)11log(2 x
)9log(5)11log(2)11log(
x
387.0x
xx 512 9log11log
Jeff Bivin -- LZHS
Solve for x
)5log()1()3log()2( xx
)5log(1)5log()3log(2)3log( xx
)3log(2)5log()5log()3log( xx
)3log(2)5log()5log()3log( x
)5log()3log(
)3log(2)5log(x
12 5log3log xx
151.1xJeff Bivin -- LZHS
Solve for x
)3log()89()7log()23( xx
)3log(8)3log(9)7log(2)7log(3 xx
)3log(8)7log(2)3log(9)7log(3 xx
)3log(8)7log(2)3log(9)7log(3 x )3log(9)7log(3
)3log(8)7log(2x
209.1x
8923 3log7log xx
Jeff Bivin -- LZHS
Solve for x
)ln()23()15ln( ex
23)15ln( x
x32)15ln(
x3
2)15ln(
x569.1
23ln15ln xe
1
Jeff Bivin -- LZHS
Solve for x
)ln()65()ln( 37 ex
65)ln( 37 x
x56)ln( 37
x5
6)ln( 37
x 031.1
1
6537 xe
6537 lnln xe
Jeff Bivin -- LZHS
)2log()13()log( 75 x
Solve for x
)2log(3)2log()log( 75 x
x
)2log(3
)2log()log( 75
x172.0
13275 x
1375 2loglog x
)2log(1)2log(3)log(75 x
Jeff Bivin -- LZHS
