I A 9-2 I
I A 9-2 I
Name
Date
;;
Period
_
'1.,3Worksheet - Properties of Logarithms
Expand a~uch
11. 2 In 6 - In 3
as possible using the properties of logs.
1. 1092(15fx2)
2. 1093
\(\ ~=
+
'n l~
~
\O~
1
14. -(2IogbM-logbN-~
13. log A - 2 log B + 3 log C
A-c.?>
\ ~~
p-
3. logs (ley)2
3
H:l/~
~ \°3 b
_.
lI_
4. InVYi
..[i_
- ~ \c:js-=t
J \~~- \00
15. log
1T+
1
16. log22 + log2 6 - - log29
2 log r
2
2-
\n) ~
\~"f
5. In (xe
N
Ib5;S-:X \D~"?,'j..-\O~~~
\DJ~lb +~\~~~
~l~). t ~ kxj51
12. logM-3log
-
~Jd-. -:0 -
\0:;.
tt ~ )- Jd-
X
\ 't"\
)
6.I{i)
Evaluate without using a calculator.
\~\()y..
y.. -\- '"
17. In e2
18. In
7.
4 \t1j X -
\tXj lx-J-)
8.
,Jx ..fX+1]
L (x-2)2
3
\~ }(.t~09ext l)-~\~()(~J
Write each expression as a rationas number or as a single logarithm.
\~~'~.4
\~ ~L\-
l6
I.
,
22. 101Og3
+ 1094
I
Co
23. e1n.f2
~
\bCJlo
~
21. 102ioge
20. 10¥
1
10. 210g.9+109.5
\~J>D5
,
Fe
f
- , 3~
,..
24. 33iogs 2+iog38
I
tl1
I;;L
~5. 10945 - log.60
I
t
9. log 2 + log 3 + log 4
19. In
e
-\
~
x·
logx_2
!
~Y
t
1t::l4 U;; -=- -.:l...
.,
~~
Name
__
Date,
_
Period
EJ
_
25. 1og3(2~)=
WORKSHEET
3. 10Q,M = N
-1.\
0;, = ~5
2. log _1_
=-4
10000
X -:::M
4. In e
'eN
\ LD
~
5. log168 = %
<6
=
6. In 1 = 0
ll
\~~,=-3
\0 <0 ~ -y..
9. e"=8
l ~-J..'2..
11. 10~=NDfj
_,
\0
I
-=
I/JDOCXJ
\-J1o e"::-l\'
8. x"= e
\~
10. 81%= 3
~
~~ V'-l
~~-~
1
12. 6-2 = 36
28. In(-10)=~
29. In e4.5
13. log 100 =
16. log39=
1
19. 10925-
5
22. In e
=
~
=~
Do not use a calculator.
14. log 10,000 =
__L
_j__
_1_
15. log 0.01 = -
18. log338=
17.logI62=L
_L
~
27. 10021 =
_D__
20. log, ~ =
=3.
21. log,V4
23. Ine2=
~
24. InJe =
;;2..
~
=_i_
...L
~
=-±-S
30.logO=
Y)
)CL..-
Evaluate each logarithm using the change of base formula. Round
answers to three decimal Place~pO
\
\00
31.10,,25
\~~,."J'
33. logs .35
\n q 02 ~-~~~g~
~1:9 g ~
32.10",00
\"Y\ '5
35. logs <-36)()D1-
Evaluate each logarithm.
26. log~9 = -
7
e..:::e...
e.0 -=-- I
1
Write in logarithmic form'
7. a3=64
3
- LOGARITHMS
Write in exponential !2rm.
1.log.25=x
-
EJ
~s 'o\e.-
16
36. 109251200
~
\(\\J--z...
~
~
a·
-
~
15
LJ
l:lDO _ 'XJ q
\~~'O
I
:;to3
I
I A 9-3 I
NAME
DATE
WORKSHEET
Solve.
1. 1092 (2x+1)
=3
ir
~.A
~~-r\=g
\
= 210g32
Y"2.
\
fXj~'I. ~
=2
3
)("'\
""2-
4-
~3
3109 2 (X - 1) + log
.3
;;l"~:/'J.-l)t
\-=-9
tt( "j_-\)
4
i
~91J2(3X+1)113
~.".
1'2
_A.
3-"
3'"
C>: d~'
=- L}
x-\ -= t.3
~~ 4 J -;)_
4-
~.?
3:l
-y
"f..- \ -;: J-
if
-;0.
'j..."__ d.. 'I..
I
La L\
~'1_
\
A 9-3
J
y;.-?\if'
J
_ '" +\ '5 ~ -:;:\ 00
'y..,-'2-{\OX.-\CO:;'u
('I- -\-ao'; ('f. -:S-y:_o
.. ~~O~
:----::-
+)
-t '-t'l- -
---
- ~-\ tv
'I.. J,;?:>.-
;f
-.IOg.(X
_ \
( i.:::.L
')L+3 - ~~~C,p
~
=2
(x + 4 )
~
\?- -::.-/ -+ ?-.'/. - 3;,
~'L.~C1
i =- 5.:A
'n
)(.
_ '-\
~y..'f..-\
'#.-
J o,
~
-e,
~¥.\-
11 . log • (x - 2) - log.(x + 3) = log. (1)
x -
100 ~('l-t-,'5)-=- ~
leA
~
~
CQ
= In
\() xCy.-;» -;
~
\~~~=\O~54
6. 109,X + log (x + 15 )
In x + In (x - 2)
10.
'3~ -=- it; 3
"2-
:::
t:
(3)(115 =.
~
@
2-5
=-l,
1
'J.-=
('/..
--\)~
~6
~
=2
L\- ~ 'f.
9
0'J. )It
~~ :! e
(3
-
8. 1og3(X _1)2
L J:X f
A
4. 2 logs)( = 310g5 4
C~~ Itcl
5.
2. Iog3( i + 1 )
3A
X::2.;-
=2
3'#. t- \
(JV~=(Ur)2-
\
EQUATIONS
G~~~~
:l
~
2"1093l'
7. log. 4
"I.. -= t. IS
~~l
3.
- LOGARITHMIC
-:
PERIOD __
"J..~ -::. <6
"::;
1
_
~t"4)
t- y
-=-
0
vi=«>
I
I A 9-31
Evaluate each logarithm.
12. log33
Do not use a calculator.
~
a\..-
13. Iog( - 1
~..
~
64
I _ ";
- ~
15.
14. log (-0.01,)
l\O-t-
210020.25
b
~'S<s\b\e.
;(5
Expand as much as possible using the properties of logs.
16. log~
17. log
\X2 + 1)
\ OS to -lo:/'t."\\)
~2.;mr
'
81D3m + d- \~
\D \03
rY1
(Y'\
(
~
I.>
Write each expression as a rational number or as a single logarithm.
1
'
19. -log8+I095
3
18. log 3 -log 12 + log 6
109
~
10:1 ~
0
Ip
I
lo~ ~ CI:5
\03 10
1-
J
~
I A9-4 I
Name
Date
Period_
7. 1.2" = (O.srx
WORKSHEET - USING LOGS TO SOLVE
EXPONENTIAL EQUATIONS
'f.
8.
\n \.? ~ -'/.,.Vn O.S
(\-X)~1l" -:::~\o e
){ \ cd 1'2. -t X \ t"\ 0, ~;:-D
~-~'\Y:=)(.
1. 22x+l= 4 ~
...{~td: ~
I
a~-+ \ :::d~~::;
lOCj ~
'1.
7:.
<,
\
~
y.._-=,
E~\
3. 2x= 10
X
2~3?':
=:
\~?
~S.3~
=
'1Y
LJ
-'" \r\~ ~ \(\ \.'J- .;
V_
\V\q;
,,_
!\.Ii -
\n ct r-: •
r(J
U\)
11.
;:;t3')104
~-Z})\~
e.
(~r.,'-'
3_~~\'"~.,~\(\L\ t "0\(\ '\ :-J.l~ ~ -;.(\ ~'I.) IIXj "
I\')":> - 2>\('1 '-\ -=- ~ \ ()L\ t d.'f..\'" 'J. \~ ~ "" \CIj"1 -)<. \~ '1
, 1"\ '0_3\(\ \-\<: )(01"1L\ +').\I')~ 'J.. \0<;)~k ').\t:C:J'l~I~ '1
\~?_3\""i -= 'f..
~ ( \~.\- ~ I')::> \b~-'
\VI "t\- ~I,,?
'f..;;. \00.-.
~ I
_ • <t.5l\- ~X.
~ "3lst\ttS' ' '?;P
\ 1'\
Q\
~
1-
'f.. =- \.s)?Jb..
1'1. !)O
,02- NUl
'7
I
13
~
_'J... ~
,)
3e"'"-::~~
::T51)
-"J..
.,.,
,(//..;;,f()
12. -14 + 3eX = 11
\
\~e~= ~
\f\e
3\(\ ~
l
~oo
S
-
31- \n a= \n ~
'X -::: \ n~
I?O
1\
cJ.-
400 =350
--.:x
he
\te~
=8
., '3'1-_
B
~
tD~~ -=:. \()
\(\L1-\1' \'1"2-
10. 5(2·X)
9.~'02x=~
.D'O~~
\ (\~
:::...J.
---s;
,t1l--'
12
-X-="
1\ -:::
~ -\fr~
'\\" :;:-). (l+ W)
~
G::--±.?-\
4. 8 -x
\O<j \D
34
= eX
",1-.
e'f.. _1 :A'f:>
n
X
..\...
\
\0 4
"1...-:; - \'(1-\
i ~\,
qL\:~
-\~
-=-
~:,
\n ~ ~ ~,llO
ANSWERS: 1. Y:. 2. 2, -2 3. 3.322 4. -.088 5. -.854 6. 1.356 7. 0 8.. 534 9. 20.273 10.. 226
11.1.946 12.2.120 13.9 14.2 15.2 16.8 17.4 18.,10 19.f(x)=lo93(-(X-4))-5 20.f(x)=4!<+'+1
>-.
~l
[A 9=6]
Name
_
Date
__
Period
3. The half-life of a certain radioactive substance is 14 days and there are 6.58 g
present initially. When will there be 1 gram remaining?
EXPONENTIAL AND LOGARITHMIC APPLICATIONS
1. The number of bacteria in a petri dish culture grows continuousty.and doubles
every 3 hours. Initially there are 2500 bacteria present.
a) Write an equation to model the continuous growth of the bac .
:::: ~
3
.
~lt)= ~5COe.,
~~e0"'"'
~~\
>,
t~3\t
~CD-::::'~5COe""?'
4. Find the principal needed to ha~e a balance of $1000 after 2 years at 6%
compounded monthly.
~ -=- ~
w
5. When will an investment of $1000 be worth $1500 if the balance is compounded
continuously at 5.25%?
,
2. a) The population of a ~Rlall tow" In me year 1890 was 6250. Assuming it
iQ.CLeasedat the rate of.3..ZS!'A>per year, what was the population in 1915?
(Round to the nearest person.)
t_
~
-t
A(-t:)~ (oa50(l +,01>15)
~t1-$)::: 1o£.6L)
(\-t-
~,
.o?:>1s1 z:
b) In1940?
V\(_:Sb)~ ~0l'tC) (\
\
-
~'~f
15lc~q
~~()
\-,t)~1 ') z:
E
313<g\
-l
(\IO~tS)t
t jlA-1.0375""
~
~:t
::
__4
~?~
•
7;>
\<6.<6
I
~t
\~qo
~,~
4t
(\r) \
6. You will be buying a new car for $15,000 in 3 years. How much money should
you,invest now at 5% compounded continuously, so that you will have enough to
buy the car?
c) What year would the population be expected to double according to the
model?
\z Sex,:)-=. lo~S-D(\ .03,5)
(
()
,.
)
'.
~,
~
~
..
,
Name
f.,
"
WORKSHEET
"I-
1. ~= 1252~t~
6=-6
'I.. -::- (0 \-~
_
Period
)L
C3\
3. 21092(4x)-1 og2 x= 3
)_~
(4~f_ e
yt
"'Z.
X:=\1?'
ti" .1.
z:
1'i-- \ ~
2. log. x = 3 -log. (20 - x)
,-\X~'l )(CZ..o -)G) "4'?"""---;o. \ LX (w-~) -::.
to4\ 'k"'l~&..\
g,o'7-..- X d--~. (o4
')(
?.....
9D y.. -\-co- L\ 0
is. -\lQJCf-~ J -=- c)
"C.-
1.
'2.)£ - \ ~ -
2'1. ~ '2
i
7 ~-;.\
~~
IrA~ ~
)(.·t 3::
)(.\(\~
~-')l..\no~3
~/\
\'1.\:: q
~
2x
10
"z.i
3e'
62-x
6. -=15
7
:r'1-
'\.-?"".
13",32-'
\'" LO ::;Wb5"
'"
-4~
= 10
e-L_IO
«: 3~)
:JD:;')(
7'\
-:0
~
..~~~
(g - ~ (?'1t-.el'-':;
e""'-::;.S
)G ~
-,;-
"'~
\0 5'
•
:i~-.i
:~
~:
'i~ I. {J:f(
12.
~e2x
- e2x
,~
~ \fI~-'-\\(\5~)_ \0'5 -\).\,,3
(p
-
_ \ (\ \D-::>
~
~ ,;;
)l
e.-do
f'
~'1;,.\;::.D
'Z.-
\
JfuJ
14. xlnx=x
G-~)\O~ ~ l'j.tl\)\n5 .. y.. \() 'i - x:;: 0
'A. (\(1 ~ - ,) ~o
a \0 '3 - \ \"'~:: ~ \o':)t'-\
1)(0 \(l~-\:O
0. \D5_
\nIO'S - J_
\n lo
=0
e2~ (x=-t)::: (.)
Cf:1.()14J
\~-Qm2»
\./ "/ ~
x~;;'_)L
r-:
9-tb-~- Y-\~P;Q0-:f'f-
~ ~ \(\-1.
'X:::
1
<O~
11.2e2x-7e'+3=0
\V\
9-5
t'V.
:=
~<t 4~
)(~~~~ttl1.
(~~- I) (e~- 3) ::: 0
~e -::\ \(\e~ :::~3
\
\'Ae~ -:;~
"'J- -::;.\n '3
~_ ',( -=
A
x...
-y:,
::=
.}...1:tJ .;! I;O-~_l~
v.(~ \n:>\-:- ~
I..
J
~.) ~
'(__jj
9~~
\CX(4;2,
J,.~
4. log31xl = 2
(?- X) \n lo ::-\n \05
~
60 log. 2 - X log. 2 -- x
8.
\2~-1 \ /
ll,q ).
\~=!\
s(_Xj1C7 )~
1
7. log~12x - ~ = 1
- MORE PRACTICE SOLVING
LOG EQUATIONS
~
'" =--B
_
\~t.f 'I.. -r lC~4l20-'A ~ 3
-~)L~.b
t ~d-~?- )(.
1A 9-51
Date
\?~-.~q7J
2 \1\ ~.-l\\() j =- ~l\~~\~
~ \{) ~ -L\\f\S ~ 'f._
\ (\~\- \" ?:>
- ~5lRlp~)<.
\ () y.. -;:. \
e:-
..e:"1.
2Y:=-~J
~~
~m
~~
m:
'm
"'-CD
~;
..,-cn
_0
CD CD
~.,..:
NO~
(")
, N
.N
~..,..
0
(J)e->
a:: .
~cn
~~
«.0
[A9-6!
7. StrontJum-90 is a radioactive material that decays according to the equation
A Ao e-O·02441,where t is in terms of years. What is the half-life of strontium-90?
=
8. The US population, in millions, is growing according to the formula P = 263eo.009I
where t is years since 1995. In what year would the population be expected to
reach 300 million?
9. A fossilized leaf is found to contain 10 micrograms of carbon-14 whereas a leaf
of this type normally contains about 13.5 micrograms of carbon-14. Estimate the
age of the fossilized leaf. (The half-life of carbon 14 is 5730 years. Round to the
nearest year.)
Answers: 1. a) A(t) = 25OOe·2311 b) 15.969 hours 2. a) 15,689 people b) 39,381 people c) 1908
3. 38.053 days 4. $887.19 5. 7.723 years 6. $12,910.62 7. 28.408 years 8. 2009 9. 2481 years
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