Worksheet: Logarithmic Function PDF

LOGARITHMS EXAM QUESTIONS

n. Page 15. Created by T. Madas. Created by T. Madas. Question 26 (***+). Solve each of the following exponential equations giving the final answers correct to.

Worksheet: Logarithmic Function

z8. Page 2. 4. Write the following equalities in exponential form. (1) log3 81 = 4. (2) log7 7 = 1. (3)

EXPONENTIALS & LOGARITHMS

ln9 ln28 ln12 ln49 x x. +. = + giving the answer as an exact fraction. 1. 2 x = Question 14 (***). Rearrange each of the following equations for x .

Higher Mathematics EXPONENTIALS & LOGARITHMS

It is common in applications to find an exponential relationship between exponential and logarithmic functions. EXAMPLES. 1. Shown below is the graph ...

Exponential and Logarithmic Equations

Solution (a): To solve this equation we will use the guidelines for solving exponential equations given above. Step 1: The first step in solving an exponential

Logarithmic Functions

with a > 0 and a ≠ 1 is a one-to-one function by the Horizontal Line. Test and therefore has an inverse function. The inverse function of the exponential.

Chapter 6: Exponential and Logarithmic Functions Selected

Chapter 6: Exponential and Logarithmic Functions. Selected Solutions to Odd Problems. 124. Section 6.3 To answer these questions I need to know what each part.

13EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Example. 1 page 928

Exponential and Logarithmic Functions

extraneous solutions with logarithmic equations it is wise to determine the domain Chapter Test Prep Videos include step-by-step solutions to all chapter ...

Derivative of exponential and logarithmic functions

functions which involve exponentials or logarithms. Example. Differentiate loge (x2 + 3x + 1). Solution. We solve this by using the chain rule and our ...

Worksheet: Logarithmic Function

Write the following equalities in exponential form. (1) log3 81 = 4. (2) log7 7 = 1. (3) log1. 2.

LOGARITHMS EXAM QUESTIONS

Created by T. Madas. Question 26 (***+). Solve each of the following exponential equations giving the final answers correct to. 3 significant figures.

Sample Exponential and Logarithm Problems 1 Exponential Problems

Sample Exponential and Logarithm Problems Solution: Note that ... Using the power of a power property of exponential functions we can multiply the ...

Exponential and Logarithmic Functions

Chapter 10 is devoted to the study exponential and logarithmic functions. 10.7 Logarithmic and Exponential Equations ... Skill Practice Answers.

Exponential and Logarithmic Functions

The correspondence from x to f1g1x22 is called a composite function f ? g. 1. 4.1 Composite Functions. Now Work the 'Are You Prepared?' problems on page 286.

Higher Mathematics EXPONENTIALS & LOGARITHMS

4 Exponentials and Logarithms to the Base e Remember that the graph of an exponential function ( ) ... expressing your answer in the form log log.

Solving Exponential & Logarithmic Equations

Original Equation. Rewrite with like bases. Property of exponential equations. Subtract 2 from both sides. The solution is 1. Check this in the original

DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC

find second order derivative of a function. l state Rolle's Theorem and Lagrange's Mean Value Theorem; and l test the

derivative-of-exponential-and-logarithmic-functions.pdf

Differentiate loge (x2 + 3x + 1). Solution. We solve this by using the chain rule and our knowledge of the derivative of loge x. d dx.

Integrals Exponentials and Logarithms Techniques of Integration

What is dxldy when x is the logarithm logby? Thpse questions are closely related because bx and logby are inverse functions. If one slope can be found

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_dbs14.pdfusesText_20

[PDF] Worksheet: Logarithmic Function

Vanier College Sec V Mathematics

1. Find the value ofy.

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

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

2. Evaluate.

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

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

4(4) logxyz

4(11) logxrpx

4. Write the following equalities in exponential form.

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

5. Write the following equalities in logarithmic form.

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

6. True or False?

7. Solve the following logarithmic equations.

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

8. Prove the following statements.

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

10. Solve the following equations.

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

2x5x12 = 0 (8)e2x2ex= 15

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

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

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

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

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

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

ANSWERS

1. (1) 2

2. (1) 0

4. (1) 3

5. (1) log

864 = 2

1010000 = 3

7. (1)S=fe3g

8. (1)

2logx4logb

10. (1)S=f2:402g

11. (1)

Dom(f) =]0;+1[

R(f) =R

Zeros: 1

Y-intercept: None

Variation:

Extremums: Max: None, Min: None

R(f) =R

Zeros:1

Y-intercept: None

Variation:

Extremums: Max: None, Min: None

Dom(f) =]3;+1[

R(f) =R

Zeros: 4

Y-intercept: None

Variation:

Extremums: Max: None, Min: None

R(f) =R

Zeros: 2

Y-intercept:2

Variation:

Extremums: Max: None, Min: None

Dom(f) =]1;+1[

R(f) =R

Zeros: 0

Y-intercept: 0

Variation:

Extremums: Max: None, Min: None

R(f) =R

Zeros:1

Y-intercept: 2ln32

Variation:

Extremums: Max: None, Min: None

Dom(f) =]2;+1[

R(f) =R

Zeros:1:5

Y-intercept: ln4

Variation:

Extremums: Max: None, Min: None

R(f) =R