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Vanier College Sec V Mathematics

Department of Mathematics 201-015-50Worksheet: Logarithmic Function

1. Find the value ofy.

(1) log

525 =y(2) log31 =y(3) log164 =y(4) log218

=y (5) log

51 =y(6) log28 =y(7) log717

=y(8) log319 =y (9) log y32 = 5 (10) log9y=12 (11) log418 =y(12) log9181 =y

2. Evaluate.

(1) log

31 (2) log44 (3) log773(4)blogb3(3) log2553(4) 16log48

3. Write the following expressions in terms of logs ofx,yandz.

(1) logx2y(2) logx3y2z (3) logpx 3py 2z

4(4) logxyz

(5) log xyz (6) logxy 2 (7) log(xy)13 (8) logxpz (9) log 3px 3 pyz (10) log4rx 3y2z

4(11) logxrpx

z (12) logrxy 2z 8

4. Write the following equalities in exponential form.

(1) log

381 = 4 (2) log77 = 1 (3) log12

18 = 3 (4) log31 = 0 (5) log 4164
=3 (6) log6136 =2 (7) logxy=z(8) logmn=12

5. Write the following equalities in logarithmic form.

(1) 8

2= 64 (2) 103= 10000 (3) 42=116

(4) 34=181 (5) 12 5 = 32 (6)13 3 = 27 (7)x2z=y(8)px=y

6. True or False?

(1) log xy 3 = logx3logy(2) log(ab) = logalogb(3) logxk=klogx (4) (loga)(logb) = log(a+b) (5)logalogb= log(ab) (6) (lna)k=klna (7) log aaa=a(8)ln1x = lnx(9) lnpx xk= 2k

7. Solve the following logarithmic equations.

(1) lnx=3 (2) log(3x2) = 2 (3) 2logx= log2 + log(3x4) (4) logx+ log(x1) = log(4x) (5) log

3(x+ 25)log3(x1) = 3 (6) log9(x5) + log9(x+ 3) = 1

(7) logx+ log(x3) = 1 (8) log2(x2) + log2(x+ 1) = 2

8. Prove the following statements.

(1) log pb x= 2logbx(2) log1pb px=logbx(3) logb4x2= logbpx

9. Given that log2 =x, log3 =yand log7 =z, express the following expressions

in terms ofx,y, andz. (1) log12 (2) log200 (3) log 143
(4) log0:3 (5) log1:5 (6) log10:5 (7) log15 (8) log60007

10. Solve the following equations.

(1) 3 x2 = 12 (2) 31x= 2 (3) 4 x= 5x+1(4) 61x= 10x (5) 3

2x+1= 2x2(6)101 +ex= 2

(7) 5

2x5x12 = 0 (8)e2x2ex= 15

11. Draw the graph of each of the following logarithmic functions, and analyze each

of them completely. (1)f(x) = logx(2)f(x) = logx (3)f(x) =log(x3) (4)f(x) =2log3(3x) (5)f(x) =ln(x+ 1) (6)f(x) = 2ln12 (x+ 3) (7)f(x) = ln(2x+ 4) (8)f(x) =2ln(3x+ 6)

12. Find the inverse of each of the following functions.

(1)f(x) = log2(x3)5 (2)f(x) = 3log3(x+ 3) + 1 (3)f(x) =2log2(x1) + 2 (4)f(x) =ln(12x) + 1 (5)f(x) = 2x3 (6)f(x) = 233x1 (7)f(x) =5ex+ 2 (8)f(x) = 12e2x

13. 15 000$ is invested in an account that yeilds 5% interest per year. After how

many years will the account be worth 91 221.04$ if the interest is compounded yearly?

14. 8 000$ is invested in an account that yeilds 6% interest per year. After how

many years will the account be worth 13709.60$ if the interest is compounded monthly?

15. Starting at the age of 40, an average man loses 5% of his hair every year. At

what age should an average man expect to have half his hair left?

16. A bacteria culture starts with 10 00 bacteria and the number doubles every 40

minutes. (a) Find a formula for the number of bacteria at time t. (b) Find the number of bacteria after one hour. (c) After how many minutes will there be 50 000 bacteria?

ANSWERS

1. (1) 2

(2) 0 (3) 12 (4)3 (5) 0 (6) 3 (7)1 (8)2 (9) 2 (10) 13 (11)32 (12)2

2. (1) 0

(2) 1 (3) 3 (4) 3 (5) 32
(6) 643. (1) 2logx+ logy (2) 3logx+ 2logylogz (3) 12 logx+23 logy4logz (4) logx+ logy+ logz (5) logxlogylogz (6) 2logx2logy (7) 13 logx+13 logy (8) logx+12 logz (9) 13 (logxlogylogz) (10) 14 logx+12 logylogz (11) 54
logx12 logz (12) 12 logx+ logy4logz

4. (1) 3

4= 81 (2) 7 1= 7 (3) 12 3 =18 (4) 3 0= 1 (5) 4 3=164 (6) 6 2=136 (7)xz=y (8)m12 =n

5. (1) log

864 = 2

(2) log

1010000 = 3

(3) log 4116
=2 (4) log 3181
=4 (5) log 12 32 =5
(6) log 13 27 =3
(7) log xy= 2z (8) log xy=126. (1) True (2) False (3) True (4) False (5) False (6) False (7) True (8) True

7. (1)S=fe3g

(2)S=f34g (3)S=f2;4g (4)S=f5g (5)S=f2g (6)S=f6g (7)S=f5g (8)S=f3g

8. (1)

log pb x= 2logbx log pb x=logxlog pb logx1 2 logb = 2 logxlogb = 2log bx(2) log 1pb px=logbx log 1pb px=logpx log 1pb 12 logx 12 logb =logxlogb =logbx(3) log b4x2= logbpx log b4x2=logx2logb4

2logx4logb

12 logxlogb 12 logbx = log bpx9. (1) 2x+y (2)x+ 2 (3)xy+z (4)y1 (5)yx (6)y+zx (7) 1x+y (8)x+yz+ 3

10. (1)S=f2:402g

(2)S=f0:369g (3)S=f7:213g (4)S=f0:438g (5)S=f1:652g (6)S=fln4g (7)S=flog54g (8)S=fln5g

11. (1)

Dom(f) =]0;+1[

R(f) =R

Zeros: 1

Y-intercept: None

Variation:

f(x)%ifx2]0;+1[ f(x)&ifx2 ;

Extremums: Max: None, Min: None

Sign: f(x)0 ifx2]0;1] f(x)0 ifx2[1;+1[(2)Dom(f) =] 1;0[

R(f) =R

Zeros:1

Y-intercept: None

Variation:

f(x)%ifx2 ; f(x)&ifx2] 1;0[

Extremums: Max: None, Min: None

Sign: f(x)0 ifx2] 1;1] f(x)0 ifx2[1;0[ (3)

Dom(f) =]3;+1[

R(f) =R

Zeros: 4

Y-intercept: None

Variation:

f(x)%ifx2 ; f(x)&ifx2]3;+1[

Extremums: Max: None, Min: None

Sign: f(x)0 ifx2]3;4] f(x)0 ifx2[4;+1[(4)Dom(f) =] 1;3[

R(f) =R

Zeros: 2

Y-intercept:2

Variation:

f(x)%ifx2] 1;3[ f(x)&ifx2 ;

Extremums: Max: None, Min: None

Sign: f(x)0 ifx2]2;3[ f(x)0 ifx2] 1;2[ (5)

Dom(f) =]1;+1[

R(f) =R

Zeros: 0

Y-intercept: 0

Variation:

f(x)%ifx2 ; f(x)&ifx2]1;+1[

Extremums: Max: None, Min: None

Sign: f(x)0 ifx2]1;0[ f(x)0 ifx2]0;+1[(6)Dom(f) = ]3;+1[

R(f) =R

Zeros:1

Y-intercept: 2ln32

Variation:

f(x)%ifx2]3;+1[ f(x)&ifx2 ;

Extremums: Max: None, Min: None

Sign: f(x)0 ifx2[1;+1[ f(x)0 ifx2]3;1] (7)

Dom(f) =]2;+1[

R(f) =R

Zeros:1:5

Y-intercept: ln4

Variation:

f(x)%ifx2]2;+1[ f(x)&ifx2 ;

Extremums: Max: None, Min: None

Sign: f(x)0 ifx2[1:5;+1[ f(x)0 ifx2]2;1:5](8)Dom(f) =] 1;2[

R(f) =R

Zeros:53

Y-intercept:2ln6

Variation:

f(x)%ifx2] 1;2[ f(x)&ifx2 ;

Extremums: Max: None, Min: None

Sign: f(x)0 ifx2[53 ;2[ f(x)0 ifx2] 1;53

12. (1)f1(x) = 2x+5+ 3

(2)f1(x) = 3x13 3 (3)f1(x) =12 102x2
+ 1 (4)f1(x) =12 e1x+12 (5)f1(x) = log2(x+ 3) (6)f1(x) =13 log3x+ 12 (7)f1(x) =ln2x5 (8)f1(x) =12 ln1x2

13. 37 years.

14. 9 years.

15. 53 years old.

16. (a)f(t) = 1000021:5t. Wheretis

the number of hours. (b) 28 284 bacteria. (c) 92.88 minutes.quotesdbs_dbs6.pdfusesText_12