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Write the following equalities in exponential form. (1) log3 81 = 4. (2) log7 7 = 1. (3) log1. 2.
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Worksheet: Logarithmic Function
Write the following equalities in exponential form. (1) log3 81 = 4. (2) log7 7 = 1. (3) log1. 2.
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Worksheet2:7LogarithmsandExponentials
Section1Logarithms
Denition:
meansthat ify=logbxthen x=by y=logbxandx=byareequivalentstatements.Example1
:Iflog4x=2then x=42 x=16Example2
:Wehave25=52.Thenlog525=2.Example3
:Iflog9x=12then x=91 2 x=p 9 x=3Example4:Iflog2y3=4then
y 3=24 y 3=16 y=163 y=48Exercises:
1.Writethefollowinginexponentialform:
(a)log3x=9 (b)log28=x (c)log327=x(d)log4x=3 (e)log2y=5 (f)log5y=22.Writethefollowinginlogarithmform:
(a)y=34 (b)27=3x (c)m=42(d)y=35 (e)32=x5 (f)64=4x3.Solvethefollowing:
(a)log3x=4 (b)logm81=4 (c)logx1000=3(d)log2x 2=5 (e)log3y=5 (f)log24x=5Section2PropertiesofLogs
log bmn=logbm+logbn log bm n=logbmlogbn log bma=alogbm log bm=logbnifandonlyifm=n Page2Example1
log bxy z=logbxylogbz =logbx+logbylogbzExample2
log55p=plog55
=p1 =pExample3
log2(8x)1
3=13log28x
13[log28+log2x]
13[3+log2x]
=1+13log2x
Example4
:Findxif2logb5+1
2logb9logb3=logbx
log b52+logb912logb3=logbx
log b25+logb3logb3=logbx log b25=logbx x=25 Page3Example5:
log 28x32y=log28x3log22y
=log28+log2x3[log22+log2y] =3+3log2x[1+log2y] =3+3log2x1log2y =2+3log2xlog2yExercises:
(a)log2xylog2x2 (b)log28x2 y+log22xy (c)log39xy2log327xy (d)log4(xy)3log4xy (e)log39x4log3(3x)22.Findxif:
(a)2logb4+logb5logb10=logbx (b)logb30logb52=logbx (c)logb8+logbx2=logbx (d)logb(x+2)logb4=logb3x (e)logb(x1)+logb3=logbx lnx=logex Page4 e y=xandlnx=yExample1
:elogea=aExample2
:ealogex=elogexa=xaExample3
log ee2y=2ylogee =2yExample4
:logex25=2logexloge5Exercises:
1.Useyourcalculatortondthefollowing:
(a)ln1:4 (b)ln0:872 (c)ln6:43:8 10 (d)e0:62 (e)e3:8(f)(e0:24)2 (g)e1:4e0:8 (h)6e4:1 (i) e8:2 1068(j)e2:4e6:1(8+ln2)
2.Simplifythefollowing
(a)logx2logxy+4logy (b)ln(8x)12+ln4x2ln(16x)12
(c)e6e6(d)12e76e2 (e)lne2 (f)ln(e2lne3) Page53.Findxineachofthefollowing:
(a)lnx=2:7 (b)ln(x+1)=1:86 (c)x=e9:8e7:6 (d)6:27=ex (e)4:12=e2x Page6Exercises2.7LogarithmsandExponentials
1.Evaluate
(a)log101000 (b)log41 (c)log327 (d)log21 4 (e)logaax2.Solveforx
(a)log4x=2 (b)log1 3x=4 (c)log10(2x+1)=2 (d)log264=x (e)logb81=43.(a)Useloglawstosolvelog3x=log37+log33.
i.log1024ii.log1098iii.log10720
ittakeforthepopulationtoreachonemillion?E=1:741019101:44M
Howmanyjouleswerereleased?
Page7Answers2.7
Section1
1.(a)x=39
(b)8=2x(c)27=3x (d)x=43(e)y=25 (f)y=522.(a)4=log3y
(b)x=log327(c)2=log4m (d)5=log3y(e)5=logx32 (f)x=log4643.(a)81
(b)3(c)10 (d)64(e)243 (f)8Section2
1.(a)log2y
x (b)4+3log2x (c)log3y1(d)2log4(xy) (e)02.(a)8(b)6
5(c)18(d)211(e)112
Section3
1.(a)0.34
(b)0:14(c)0.89 (d)1.86(e)44.70 (f)1.62(g)9.03 (h)0.10(i)3.41 (j)4.652.(a)lnxy3
(b) 12ln8+2lnx(c)1
(d)2e5(e)2 (f)2+ln33.(a)14:88
(b)5:42(c)9.03 (d)1:84(e)0:71 Page8Exercises2.7
1.(a)3(b)0(c)3(d)2(e)x
2.(a)16(b)1
81(c)49.5or992(d)6(e)3
3.(a)21
(b)2 (c)i.x+yii.2yxiii.2y+x+14.(a)4.6054months
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