Pre-Calculus
Chapter 3(3.1-3.5)
Study Guide
Topics to Consider:
. Solve any exponentialequation
. Solve any logarithmic equation
. Evaluate a logarithm in any base
. Evaluate a logarithm using changeof base
' Expand an expressionusing the propertiesof
logarithms
' Condensean expressionusing the properties
of logarithms
. Write the equationof an exponential
function from a table, graph, or a situation
' Graph a logarithmic function along with any
transformations
I
I
I
I
Graph an exponentialfunction along with
any transformations
Find domain, range,and asymptotesof
logarithmic and exponentialfunctions
Determine whether a function models
exponentialgrowth or exponentialdecay
APPLY exponentialsand logarithmsto a
real life situation!
Be able to do any of the aboveWITH /
WITHOUT a calculator
Review Problems: (Try to do all of thesewithout a calculator!)
rssion
Evaluate each
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2. log I ooo{}
L l r oe,st
f {
V
toez(,25)
(9
lo" = looo
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a
l.
6 . 1 42'l4-3. -
l{
l;;
IJZ
5l-
lst
\T
5rog,+-O
E'=Glu
t
7rt-I
? t'l:
(x--)
4.loss'J-(D
z-"-*\
-A
lv
r3
-ia I\
rc(t:%ll
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2s) l\:
s,.s,o
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sr4
5
14
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t -:-----
g. )-')'
,\{3 I
fi.u-)
'--2
-\2
b
dt
T
l- =**l
b"{ lrq-
T
Expand or condenseeachlogarithmic expression
1 1 . lo g( 1 0 x 3 y )
13rogt'ry]
12. log (x2
g.,c'o\ r/Y" )
+lug'{
\Oqrur-3\t:?^
- *"---1
t\ *
\ogsoo+$.,ugx
,
_:3
3u\og1
-y---
iiL t * 3\oqx+-lo$J\
i4. 3log3x - log3(x + 1)
b3.(*)
15. log(x+ 2 ) + Iog (x - 4) - log 1 0
\oj
-'{\
-)
P(x
(u
lo
FH
16.Tlog(xf)+3log(2xll
1--1 \sqJ
,ogtW | Lytbx:r
)")
r) (8x '"
jo)
3
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round answersto the neareFtthousandth.)
Solve each equation & Show your work! (Wherenecess.ary,
lt l"srx:s
c-
3-=
v"'
\3
,it
Y.
t l . l g $ 1 z x + r',--t
0):2
t z .r c / 5 6 + 7 ) : n { 5 p x - 2 1
Bl
{-
{
Xt^l = 3 Y--L
2X +-lD = ltp
9 =Lf
2X -U
ffi_q
1 4 .l o g x * l o g x : 4
@
15.-h!.a*:7
16.ln(5)-ln(x-3):ln(1)
(b
wqx'=4
\u \ ._.r,r\
16 jr
+ i r=*-Ytr*on)-'(ort'
=*
^.F.t*)
r/
r
..x\ " i lr f ,-*.*'i'
x=))bD[:),
fr'o*.,,i*l g . 2 7 * - t- g z x
1 7 . 2 0 . 5 * * r: 4
- \_
- x t s t =t
19, 10. 4',:0
. J * =L)
'1
=O
_'.X-t _
_t . \ X
Y{E'J
f(3')
b
jr-3 = L{Y{- \
'3={ i,1-9
x;r\
(x' i)
Solve..each
.\**on-if,show
nb . r.*0
ss{!"\ ,...-,"/
your work! ([here necessary,
roundanswersto then.iie.ttrrr#'atn:)
^'?:
\d-Dft=/ll
n
i;;;^/,-'^,
-\
r r,,Ytl-}
--^iorz
ru;"-l
22.(8\*+| : 64
23.21og4+3logx:2
25. 5'-*:.20
20.2x:26
zrlne"-,
3,\o
c:*'f
'f nx* I x'+x-L=o
)
\' :-'-:--:
*
LAq\ut [$qN"L
U.
a
-u
24.ln(x+2)+lnx:1n48
.'s'::F\t
\---l
(t-x)-ln9=-[n(.zo)
z,.\^
v-3 f;i
n-\-{
Graph eacn
urapn
eachrunctron
function and
and descrtbe
describetts
its domain,
domai4 range,
ranSlfifi"dlasymptotes:
and asymptotes:
tli):'::,GXtn"q (0,,c)
3u
L
11' \\)qx
P, [1.
t
+l
r\o
27.
r(x)=
l.s.-'
\r
Shtv-rnk- "f I{
\ (hrv-rnk-of
^[u'*t
'k *-o
'r*O*na vragvr+ t
\-;x'l*i'
,,11l,n-r
).t-\
+
L{l
tlt
Tt
,
{
t
V
)
t.
.,
\
Ifr
b'rzb.[z_
B (o,Cf')
i.'cr')
V
i
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nor'(
\\{\P\ ns{v
\\'{\
B*\ \1'o
"o
X = * L--*( ,Z-)
\
\
*
n-n,
'Ln 6
Graph eachfunction and describeits domain' range' and asymptotes:
tn'Y l'
28. f(x):2-1og2x
\i
ro.\-X.
J.,\- \
J I
rr l n
l.tla
-i
'r" Y- o''/' -
Ps'{r\ x--t)
Jt
+\
4y\
eF
a jt
ii it
kq- D (s,c)
J'
xi\
.,
-^..t i, I q 1 \
-t-_t*- -r-*-e
r
,' * 'r;+<
d h none
s
]iYS
lx
It
{
Write an exponentialfilo.?ionfrom the given inform
30.
X
f(x)
-2
2.125
0
8.5
I
+.zJ
x7_
\.'L
xl'l
tl-
Ll\.3
\q= 1Jz
)=E g tD^
t7
x2_
32. The numberof rabbitsin Elkgrove doublesevery
month. Therearc 20 rabbitspresentoriginally. Expressthe
numberof rabbitsas a function of time t. Then find the
numberof rabbitsafter 1 year. When will therebe 10.000
rabbits? Show alsebraicwork to solve.
TI
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ir'*'
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wt sr,{ 1,15
o{
=le \ gz o rabbil'L
r* q$\qv
ti^\
t_9t_(,I '- r r: (.'- )
z()
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f
t " ,.''"rj'd
the populationas a function of time t. If a "town" is deemeda city with fewer than 10,000people,when will
Metroville becomeMetro-town?
P I t 1=,i,uno(, - , 0zS
{n,,tE{-.1b = +,jtr.1"tp
!_o_o_cp
a3oDO
,\^''13L{.rb]n'9-tr't
a3ooo
I11
=
=
t Jry'"3-l'{t 31.29 yqA^S
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Attention: Noteswill not be allowed,nor will any formulasbe providedfor the test!
And as alwavs.a SIGNIFICANT PORTIONbf the test will be NO CALCULATOR!
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34.b If ee) = 20 andg(7)
m,
cf
':
li.\
cfutv-ei\'"\v\1
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When wouid one be usedover
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rtl
* t ! r. . L |l
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l--l)
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t\k
1,,
r\ -'t ., {.-(\ \,:.-
-
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)
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(.-1 c\ ( t.O,\* (iv\\
\r\( l(ie(
\t
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A'rA t
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r!
J{\ " ,\ " f -I l
f rt \C
,
U. t\*rr-'t
tF ir ..c.lc{ ee ( rt- Lo n'S\tr'!\ % rrrq
35. If theequationis in logarithmicform, write it in exponeihtial
form. If the equationis in expoiiid\tiat'iblini
I\i,i\ti"iftliile,\\ r{ .
Q
write it in logarithmic form.
Log3x = y
'..\
l,
log.4 - logrc = 10
* ,,
' i .",
lbl,, r"-"1
50* = 200
,!
. .1,
,. rr ^ ../
i,,.",
x3ye= 12
't.
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36. Solvethis exponentialequationboth algebraicallyand graphically:500(0.95)*= 200(1.35)*
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37. Chemistsdefinetfre acidity Ji;f[ufiniry of u ,ubrlun"" accordingto the formula "pH = -IogfH*f" where
[H*] is the hydrogenion concentration,measuredin molesper liter. Solutionswith a pH value of less thanT are
acidic; solutionswith a pH value of greaterthanl arebasic;solutionswith a pH of 7 (such as pure water) are
neutral.
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a) Supposethat you test applejuice and find that the hydrogenion concentrationis [H*] = 0.0003.Find the pH
value and determinewhetherthejuice is basic or acidic.
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b) You test someammoniaand determinethe hydrogenion concentrationto be [H+] = 1.3 x 10-e.Find the pH
value and determinewhetherthe ammoniais basicor acidic.
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38. "Loudness"is measuredin decibels.Theformula for the loudnessof a soundis given by
"dB = l0logl I + Io ]" whereIo is the intensityof "thresholdsound",or soundthat can barely be perceived.Other
soundsare defined in terms of how many times more intensethey are than threshold sound. For instance,a cat's
purr is about 316 times as intenseas threshold sound,for a decibel rating of:
Db= 10/oglI+Iol
= l0logl (316Io)+ Iol
= I 0 l o g [ 31 6 1
= 24.9968708262....
...or 25 decibels.
prolongedexposure
Consideringthat prolonged
Considering
exposureto
to\ounds ub!u"_85iq"tb$Fan causehearingdamageor loss,and
c o n s i d e r i n g t h a ta g u n sh o tfro ma '2 2 ri @ofaboutI=( 2,5x1013) h,shouldyoufollow
the rules and wear ear protection when relaxing at the rifle range?
F.
\
\)b*
l,-
l\) \bnr
U
Db?
P":#*1
\u \-rrt(?.4 -,'.J*){r=3n BDb !.r\. o!*u"-I:
\\
LgssI:-
c.'"k:s
"rf, BgUu
39. Earthquakeintensity is measuredby the Richter scale.The formula for the Richter rating of a given quake
is given by "R = logl I + Is ]" where Io is the "thresholdquake",or movementthat can barelybe detected,and
the intensity I is given in terms of multiples of that thresholdintensity.
You have a seismographset up at home, and seethat there was an event while you were out that had an
intensity of I = 98910.Given that a heavy truck rumbling by can causea microquakewith a Richter rating of 3
or 3.5, and "moderate" quakeshave a Richter rating of 4 or more, what was likely the event that occurredwhile
vou wereout?
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40. Write a "challenging" logarithmic expressionthat can be evaluaYted
without a calculator, then confirm your
resultusingthechangeof baseformula.
Ctn'5u":ef 5 *t
y'Crr--'
r l.,
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