PDF Chapter 2. LOGARITHMS - Senior High Mathematics

[Pages:11]Chapter 2. LOGARITHMS

Date: - ? 2009

A. INTRODUCTION

At the last chapter, you have studied about Indices and Surds. Now you are coming to Logarithms. Logarithm is an invers of indices form. So Logarithms, Indices, and Surds have a strong relationship.

Look at these illustrations:

23 8

in the logarithmic form: 2log8 3

24 1 16

in the logarithmic form: 2log 1 4 16

3 log243 5 in the exponential form: 35 243

5 log 1 2 in the exponential form: 52 1

25

25

Notes: - in this chapter, we use the scientific calculator - in some countries: a log b is written as loga b - logarithms to base 10 is usually written as: log b

( not: 10log b ) - Some of exponential laws are used in this chapter

So we got the first law: a log b c b ac Where: a = base and a 1, a > 0 , b > 0

EXERCISE 1 Convert to logarithmic form or exponential form:

1. 63 216

2. 3 4 1

81

3. 1 3 8

2

4. 1 2 p

4

5. ma n

6. 2log16 4

7. 3log 3 1

1

8. 3 log 6 c

9. 7log 49 y

10. ulog v w

Logarithms, SMA PAHOA 1

B. GRAPHIC OF LOGARITHMS

Date: - ? 2009

To find graphic of logarithms, we can draw it manually or using the computer.

2.5 2

1.5 1

0.5 0 0

Graph of Log x

Graph of "2"log x

6

5

4

log x

3

2

"2"log x

1

50

100

150

200

250

0

0

10

20

30

40

50

60

Graph of 10log x

Graph of 2log x

Graph of 4 log x

3.5

3

2.5

2

1.5

1

0.5

0 0

10

20

30

40

50

60

70

80

90

100 110 120 130

EXERCISE 2

Without using a calculator, according to the graphic above, estimate the value of: (up to 2 decimal places)

1. 4 log 2

4. 4 log 36

7. 4 log ? 1,60

2. 4 log 8

5. 4 log 80

8. 4 log ? 3 ,37

3. 4 log 18

6. 4 log 104

9. 4 log ? 2,23

Logarithms, SMA PAHOA 2

C. GRAPHIC USING THE COMPUTER

Date: - ? 2009

There are many softwares we can use to plot or make graph of logarithm (and also other math graphs) such as Microsoft Excel, Graphcalc, Graphmatica, Mathematica, FXGraph, Sketchpad, Math Mechanix, etc. Each software has specific functions compare with other.

Below is the way to make the graph of log x and 2log x by using Microsoft Excel.

1. To find graph of log x :

8. To find graph of 2log x :

- Open Microsoft Excel - In column A row 1 (cell A1), type: x - At cell A2 to A9 please type the numbers you

need (exp. 1, 5, 10, 20, 50, 100, 150, and 200) - In column B row 1 (cell B1), type : log x

- at cell B2, type: =log(A2) then enter - copy-paste cell B2 to cell B9 - block column A and B, make align centered

- In column E row 1 (cell E1), type: x - At cell E2 to E9 please type the numbers you need

(exp. 1, 4, 8, 12, 16, 24, 32, 50) - In column F row 1 (cell F1) type: `2'log x (means 2log x )

- at cell F2, tipe: =log(E2)/log(2) then enter - copy paste cell F2 to cell F9 - block column A and B, make align centered - Block column E and F

- go to step 3?7

2. Block column A and B

9. Place both of graphs properly by placing them below the

3. Click: Insert Chart XY (Scatter)

two tables, resize them when needed.

Choose chart sub-type at the center (scatter with 10. At cell H1: type `your name'

data points connected by smoothed lines).

At cell H2: type `class . . . . .'

4. Click: Next twice

At cell H3: type the date `____ August 2009'

5. At "Legend" please unblocked "Show Legend"

11. Click Print Preview, make sure all graphs are shown.

6. At "Titles" please type "Graph of Log x" or later 12. Please print your work on a piece of paper.

"Graph of "2"Log x" ( means 2log x )

7. Click: Finish

ASSIGNMENT AT HOME:

Make two graphs of 5 log x and 0.5log x using Microsoft Excel. Please take ten data of "x" randomly and the "x-range" is between 1 and 100. (example: the "x-input are 1, 8, 1_, 2_, 3_, 4_, 5_, 6_, 8_, and 9_). The assignment must be collected before: ____ August 2009

Logarithms, SMA PAHOA 3

D. SIMPLE LOGARITHMIC EQUATIONS

Date: - ? 2009

We already know, that if a log b c then b a c and if b a c then a log b c

We can use that rule to do these simple equations;

1. Find the value of x if 2log x 5

2. Find the value of x if 8 log (x 5) 4

Solution: 2log x 5 x 2 5 32

Solution:

14

8 log (x 5)

4

(x 5)

(

8) 4

8

2

82

x 5 64 x 59

EXERCISE 3

Without using the calculator, find the value of x in the following:

1. 3log x 4

7. (2x3)log (x 4) 1

2. 2log (6 3x) 4

8. 3log 81 x 2

3. (9x)log 64 3

9. x log (6x 8) 2

4. 5log (18 x) 1

10. 2x log 20 2

5.

4 log

1 x3

3

11. (x1)log 1 2 64

6. 12log (3x 5) 0

12. If 4log x 2 and 2log y 3 , evaluate x y

Addition: do also the exercise from Buku Kerja, page 22-23

Logarithms, SMA PAHOA 4

E. COMMON AND NATURAL LOGARITHMS

Date: - ? 2009

1. COMMON LOGARITHMS Logarithms to base 10 (ten) are called Common Logarithms. We often write `10log' as `log' (or just `lg' in some countries). For example `10log 6' and `10log (x-1)' are abbreviated as `log 6' and `log (x-1)'" respectively. Common logarithms can be evaluated directly using a scientific calculator.

Recall that by the definition of logarithm: log b c b 10 c and sure that log 10 1

2. NATURAL LOGARITHMS Besides base 10, another important base is e. Observe that 1 1 n will approach a certain value as n becomes very large (up to infinity).

n This limiting value is denoted by e and e = 2,71828 (to 5 decimal places)

Logarithms to base e are called Natural Logarithms. Notation `elog' is often abbreviated as `ln'. So "eln 8" and `elog 2x' are written as `ln 8' and `ln 2x' respectively. Like Common Logarithms, Natural Logarithms can also evaluated using a scientific calculator.

By definition: ln a b a e b or a 2,71828 b and sure that ln e 1

Example Solve the following equations:

a) 5 x 3 60

b) e 2x 1 27

Solution:

a) 5 x 3 20 log 5 x 3 log 20 (x 3) log 5 log 20 x 3 log 20 1, 8614 log 5 x 1, 8614 3 1, 1386

b) e 2x 1 27 ln e 2x 1 ln 27 (2x 1) ln e ln 27 (2x 1) .1 ln 27 2x 1 3 , 2958 2x 3 , 2958 1 4, 2958 x 2, 1479

EXERCISE 4 Solve the equations: (up to 4 decimal places)

1. 7 x 700

x

5. e 2 888

2. 5 3 2x 65

6. e x2 555

3. 9 6 3x 1 4. e x 2 92

7. 2 x . 4 x 2009 8. 3 x 123 . 5 x

8. 6 x 3 5 x 9. 5 x 1 . 3 x 2 20 10. 2 2 x . 5 x 1 729 12. log 4x 56

13. ln 3x log 4 . log 5 14. ln x 2 36

15. ln x2 36

16. ln 5 . ln 7x 9

Logarithms, SMA PAHOA 5

F. LOGARITHM OF A NUMBER

Date: - ? 2009

Before we have calculator, scientist used a table of logarithm to find the exact value of logarithm of a

number.

Below is the example of a logarithm table:

N

0

1

2 .... 8

9

.... .... .... .... .... .... ....

21 3222 3243 3263 . . . . 3385 3404

22 3424 3444 3464 . . . . 3579 3598

23 3617 3616 3655 . . . . 3766 3784

.... .... .... .... .... .... ....

What does it mean? How do we use it? The table shows four significant figures of logarithm of a number (in column N). If that number is greater than 10 and below 100, then you must add 1 (example: log 2,3 = 0,3617 and log 22,8 = 1,3579) but if that number lies between 100 and 1000, just add 2 (example: log 212 = 2,3263), and so on.

After scientist found a scientific calculator, then we use it to find the very exact value of logarithm of any number. But all calculators are designed to base 10 only. It can find the value of log 15, log 758, log 2009, and so on because the base of them is 10. But if we want to find the value of 2log 10 , 6log 333 , 0,5log 7 we must use the logarithms laws and

log 10 log 333 log 7

change them first into

,

,

, etc.

log 2 log 6 log 0,5

Example Find the exact value of:

(up to 4 decimal places)

a) log 36 b) log 2010

c) log 0,3

d) 2log 6

Answer:

a) log 36 = 1,5563

b) log 2010 = 3,3032 c) log 0,3 = -0,5229

d) 2log 6 log 6 2,5850 log 2

EXERCISE 5

By using a scientific calculator, find the exact value of: (up to 4 decimal places)

1. log 0, 6 =

5. log 10 =

9.

1

3 log 9

13. log 20 ? log 2 =

2. log 0,06 = 3. log 45 = 4. log 4.500 =

6. log 10.000 = 7. 2log 8 8. 2log 32

10.

1

3 log 1

81

11. 5log 5

12. 5log 1 25

14. log 35 4log 7

15.

2log 16 2log 32

16.

2log 5 8log 25

=

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G. LAWS OF LOGARITHMS

Date: - ? 2009

For a 1 , a > 0 , b > 0 , d > 0 : 1. a log b c b ac 2. a log (b.c) = alog b + alog c 3. a log b alog b alog d

d

4. a log b m m . a log b 5. an log b m m . alog b

n 6. a log b . blog c alog c

7. a log b log b log a

8. a log b 1 b log a

9. a alog b = b

a log a 1

Example using laws 1-3 Evaluate the following:

a log b c b ac

1. 2log 32 ? Let 2log 32 x 2x 32 2 5 x 5

alog (b.c) = alog b + alog c

3. 2log 5 2log 0,1 ?

2log 5 2log 0,1 2log 5 0,1

2log 0,5 2log 21 1

a log b alog b alog d d

5. 3log 162 3log 2 ?

3log 162 3log 2 3log 162 2

3log 81 let 3log 81 x

3 x 81 x 4

2. 3 log 1 ? 81

let 3 log 1 x 81

3

x

1 81

1x

3

2

34

x 4 x 8 2

do exercise page 23 no 2, 3, 5, 8

4. 3log 1 3log 45 ? 5

3log 1 5

3log 45

3 log

1 5

45

3log 9 x

3x 9 x 2

6. 8log 3 ,2 8log 0,1 ?

8log

3,2

8 log

0 ,1 8 log

3,2 0 ,1

8log 32 x

8 x 32 2 3x 2 5 3x 5 x 5

3

see Matematika Inovatif page 58-59

see Matematika Inovatif page 59-60

do exercise page 27 no. 1, 2, 3, 5a

EXERCISE 6 Page 23:

2) 2log 3 x 8

Page 27: 1) 2log 24 2log 6

=

3) 1 . 3log 18 3log 2

=

8 2 . 3log 2

3) ( x1)log 32 5

5) ( 32x)log 216 3

2) 2 . log 5 3 . log 6 log 54 5a) if log 2 = 0,301 and log 3 = 0,477

=

calculate log 6 = ?

8) 2 x log 900 2

Logarithms, SMA PAHOA 7

Example using laws 4-6 Evaluate the following:

a log b m m . alog b

1. 3log 1 ? 243

3log 1 3log 3 5 243

5 . 3log 3 5 . 1 5

an log b m m . alog b n

3. 8log 1 ? 4

8log 1 23 log 22 4

2 . 2log 2 2

3

3

2. 5log 625 ? 5log 625 5log 5 4 4 . 5log 5 4 . 1 4

do exercise page 24 no 2, 4, 5, 10

1

4. 9 log 1 ? 243

1

9 log 1 32 log 35 243

5 . 3log 3 5

2

2

do exercise page 25 no. 3, 4, 6, 7

Date: - ? 2009

a log b . blog c alog c

5. 2log 7 . 7log 32 ?

2log 7 . 7log 32 2log 32

2log 2 5 5 . 2log 2

= 5.1 = 5

1

6. 4 log 6 . 6log 128 ?

1

1

4 log 6 . 6log 128 4 log 128

22 log 2 7 7 .2 log 2 3 ,5 2

Addition problems

EXERCISE 7 Page 24:

2. 2log 1 32

Page 25:

1

3. 6 log 216

Addition problems: 2log 0 ,7 . 0,7log 1 64

4. 6log 6 6

4. 10 log 0 ,01

5log 3 . 9log 125

5. 2log 1 2 8

6. if 2log 3 a then 4log 27 ?

1

6log 8 . 3log 81. 2 log 216

=

1

10. 3 log 27

1

7. if 2log 3 a then 8 log 9 3 ?

2log 0 ,7 . 0,7 5 log 1 64

Logarithms, SMA PAHOA 8

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