C7L3 Notes
Properties of Logarithims
Product Property:
1. Ioga(MN)=logaM+logaN
2. In(MN)=lnM+lnN
Examples:
1087 (3X) = 1087 3 + 1087 X
ln (20) = In 4 + In 5
Quotient Property:
3. toga(#)=1ogaM-logaN
4. In (:)=lnM-lnN
Examples:
108n (Z) =108nx-Iogn2
ln (:) =ln 8-Ina
Power Property:
5. Toga Mr = r loga M
6. |n Mr = r ln M
Examples:
log8 9X = x Iog8 9
In 3y - y ln 3
Write the expression as a sum or difference of logarithms. (expand)
1. Iogxy
LOGX + LOG3
3. Iog5tvr
LD65L+Lasgv+Life¥if
2. Inllv
Ia u! SJ,I + hi\j
4. In 3xy
fty\3+hi*+ft'\3
Write the expression as a single logarithm. (condense)
6. ln in + ln y
5. log 3 + log 5
L.Orte (3 . ¥ ``)
jj-.,...-,,.i.
L{+:){=rfra I gr
7. 1084 X + 1084 7
]iLeT}r3#F7X
8. In 5e3 + |n 9 e7
jf r! ` # F4%= pgivfu
Write the expression as a sum or difference of logarithms. (expand)
9Jog=
10. ln :
LOO I ( -LOG V
hid - th-4
12. ln±
11. log=
L0GX-LOGj-LGG€
ftyq - fu, 3 - fro ffi_
Write the expression as a single logarithm. (condense)
14. Iog7 6e2 -log7 5e8
13. log 8 -log 7
?
-, 3 -i
-.= L`
LOGl fe
Lj,G7 j-eG
15. In x -ln x4 -|n w
X,
16. Iogn 5 -logn 20 -logn 18
Lo(3o-fzo, lrd
`,
LroGrfu
LOG fr+2~
Expand.
18. In 83
17. logn 7y
I LOG„ r/
:-.;--.-.i
20. loge x7
19. logv5
r7 LOC>ex
5 L!DC) V
Can you prove the following statements using properties of logarithms?
21.i]#L;g|8o.ro3)£L/
22. 2 Iog7 3 = Iog7 9
LOG,/32 = LOcry?
AVO
Lf)G -Jq = L(j{:-/ 'q
y
23. 3 logs 3 = logs 9
L!Cfo533=LOC}5q
```: `,
24. 5 logw x8 = |ogw x4°
EZIRE
Lifewx4r)~_UorooX
ulf05gr] i tit 3;ci
//0
Write each expression as a sum and/or difference of logarithms. Express
powers as factors.
25. In=
hiexnas-fie,
#IA¥ -I
26. In (yey)
`;-'.
27. 1083
x3
28. ln (¥)
y2z*
LOG3x3-L;G-35L~LOGfa
M. (¥+1) - flit-X
£L0G3X_^dL0G3{±_UDtg:rm3%€
WL:X+]) ~ 3 ff x
Write each expression as a single logarithm.
29. 4log6X-5log6y+log6Z
30. 3log4X-1084y-210845
3
Lf)'C2{&X
LfREogf a
L06qxJ: Lored{,i'±-Lc8q5
|joG4x%-lJceqg-~Lor3rf3g
3
LC)C4
LJT±.
31. 2 log2 (x + 1) -log2 (x + 3) -Iog2 (X -1)
Loroa {X+(}L-LCJ6a(x+3) -LJOC&(X:/)
~:-:...
LJOC) 9`
(x-+3)(X+)