Chapter 4. Section 4
Page 1 of 1
C. Bellomo, revised 24-Oct-07
Section 4.4 โ Properties of Log Functions
Log Rules:
log
log
log
log ( ) log log
log log
log ( / ) log log
log
b
b
bp
bbb
bpb
bbb
p
b
bp
MN M N
Mp M
MN M N
bp
b
=
=
=
=+
=
=โ
=
1
10
Simplifying Log Expressions:
โข Example. Express log(0.2 )
x
as a sum of logs
log(0.2 ) log0.2 log
x
x=+
โข Example. Express 76
log 13
aas a difference of logs
76
log log 76 log 13
13
aaa
=โ
โข Example. Express 68
25
loga
ab
ab in terms of sums and differences
68 431/2 43 4 3
25
11 1
log log ( ) log ( ) [log log ] [4log 3log ]
22 2
aa a aa aa
ab ab ab a b a b
ab ===+=+
โข Example, simplify 55 5
2log log 3log
x
yzโโ
2
213
55 5 5 5 5 5
3
2log log 3log log log log log
x
xy z x y z yz
โโ
โโ = + + =
โข Example, simplify 32
log( ) 3log( ) 2log( / )yz x y x zโ+
32 32 3 3/2 2 2
3/2
32 3 3/2 2 2
log( ) 3log( ) 2log( / ) log( ) log( ) log( )
log( ) log
yz x y x z yz x y xz
y
yzx y xz
x
โโ โ
โโ โ
โ+=+ +
==
Calculating Values:
โข Example. Find log 9
agiven log 2 0.301, log 7 0.845 and log 11 1.041
aa a
โโ โ
log 9 log (7 2)
aa
=+, but we don't have a formula to break that down.
We could find log 22 log (2 11) log 2 log 11 0.301 1.041 1.342
aa aa
=โ
=+ =+=
โข Example. Given log 5 1.609 and log 3 1.099
bb
โโfind log 15
bb
log 15 log 3 log 5 log 1.099 1.609 1 3.708
bbbb
bb=++=++=