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EXPONENTS, RADICALS AND SCIENTIFIC NOTATION

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Exponents and Radicals

This article describes mathematical operations using exponents and radicals.

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Operations with Exponents and Powers

An ''exponent'' is a number placed above and to the right of another number, called the ''base'', which tells how many times the base is to be multiplied by itself. For example, 32means 3 x 3, 43means 4 x 4 x 4, and so on. (Raising a number to the second power is referred to as ''squaring'' while raising a number to the third power is called ''cubing''.) The importance of exponents lies in the fact that they simplify the multiplication and division processes. The product 5 x 5 x 5 x 5 can be written as 54. Multiplication of quantities with exponentsof the same baseis performed by adding the exponents. This rule may be stated as:

(xm) (xn) = x(m+n)

The product of a number raised to the mthpower and thesamenumber raised to the nthpower is just that number raised to the (m+n)th power.

(x2) (x3) = x2+3= x5

Notice that this rule only applies to multiplication. It doesnotmean that x2+x3= x5. This can be easily verified by substituting numbers in place of the x’s. Also, this process is valid only for exponents having the same base.

The division of quantities with exponents of the same base is performed by subtracting the exponent of the divisor from the exponent of the dividend. This rule may be stated as:

(xm) ÷(xn) = x(m - n)

x3÷ x2 = x3-2= x1= x

The above example could have been written as:

If the numerator and denominator are divided by x twice, the result is

In the case of multiplication and division, it is theexponentswhich are added or subtracted, not the bases.

To raise a quantity with an exponent to a power, multiply the exponent by the power.

(xm)n= xm x n= xm n

(x2)3= x2 x 3= x6

The basis of the operation in the above example is that the power 3 means ''multiply the base by itself 3 times''. So (x2)(x2)(x2) can be simplified by adding exponents. This gives x2+2+2= x6which is equivalent to (x2)3= x2x3= x6. It is sometimes easy to become confused as to when exponents are added and when they are multiplied. Remember that the exponent tells how many times the base is to be multiplied by itself. If the base itself has an exponent, that exponent is still part of the base. In the example above, the number x2is the base for the exponent 3. If a product of several factors is raised to a power, this is equivalent to raising each factor to that power. This rule may be stated as:

(xyz)n = xnynzn.

For example, consider (xyz)3. The quantity (xyz) is the base of the exponent 3 and so:

(xyz)(xyz)(xyz)= x.x.x.y.y.y.z.z.z

Each of the x’s are a common base and so x.x.x = x3. Similarly with y and z, and so (xyz)3= x3y3z3.

(2 x3y2z)4= (2)4(x3)4(y2)4(z)4= 16x12y8z4

Notice in this example that the number 2 is also raised to the fourth power, since it is within the parentheses.

A fraction is raised to a power by raising both the numerator and denominator to that power. This rule may be stated as:

Again, the quantity is the base for the exponent n. For example, compute:


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Example:

Special Exponents and Radicals

In the discussion of exponents thus far, only positive, whole numbers have been considered. There are three other types of exponents, namely, zero, negative numbers, and fractions.

How is a number having a zero exponent interpreted The rule for division is used as an aid, i.e.:

Suppose that m and n are equal. Then:

But xmxmis just equal to 1, since the numerator is equal to the denominator. Therefore, the value ofanynumber raised to the zero power is 1. xo= 1

1o= 1

3o= 1

(7943.8)o= 1

(Of course, the number zero raised to any power is always zero, i.e., 0m= 0).

Now suppose that the exponent is a negative whole number. What is the meaning of x-mAgain, the rule for division is used.

Let m have the value zero. Then:

But:

Therefore: and so any number with a negative exponent is equal to the reciprocal of that number with the same positive exponent.

Anyfactorin the numerator of a fraction may be transferred to the denominator provided the sign of the exponent is changed. Similarly, anyfactorin the denominator of a fraction may be transferred to the numerator provided the sign of the exponent is changed. For example, the following quantities are equivalent:

Note carefully the underlined wordfactor. The quantity isnotequal to a-2+b-2. The two terms in the denominator are connected by a+sign, so it is an expression. Onlyfactorsmay be transferred by changing exponents, not expressions.

We are now in a position to study fractional exponents. What is meant by the quantity (x1/2). Since the same base is used, the exponents can be added and so (x1/2)(x1/2) = x1/2+1/2= x1= x. The product of two equal quantities (x1/2) is equal to some number x and so x1/2must be thesquare rootof x. The square root of a number is the factor which when multiplied by itself gives the number as a product. The square root, then, is indicated by the fractional exponent 1/2. The exponent 1nmeans ''take the n''''th''''root'', where n is any number.

Example:

41/2= 2, since 2 x 2 = 4

361/2= 6, since 6 x 6 = 36

641/3= 4, since 4 x 4 x 4 = 64

321/5= 2, since 2 x 2 x 2 x 2 x 2 = 32

Thesymbolfor a root is theradicalsign √ and the number whose root is to be found is theradicand: nthroot =

Theindexn indicates which particular root is to be found. When the index is omitted, the square root is taken.


center

In solving problems, it is frequently necessary to identify or recognize numbers which are ''perfect squares'' (integers raised to the second power) or ''perfect cubes'' (integers raised to the third power). It is also necessary to know the square and cube roots of these numbers. The following lists squares, square roots, cubes, and cube roots which are frequently needed.

All of the fractional exponents that have been considered so far have a numerator equal to 1. The next step is to interpret what is meant by a quantity such as X¾where the exponent is the fraction ¾. We know that (Xm)n= Xmn

If m = ¼ and n = 3, then:

The fractional exponent ¾ means ''take the fourth root of a number and then raise the result to the third power''.

The fractional exponent P/R means ''take the Rthroot and raise it to the Pthpower''. All of these rules apply to negative fractional exponents as well.

Example:

Operations with Radicals

A ''radical'' is nothing more than a notation that indicates a fractional exponent. When a product of factors is raised to some power, this is the same as the product of each of the factors raised to that power.

The same operation applies to fractional exponents, that is:

This allows simplification of a radical by factoring the radicand such that an exact root of one of the factors can be taken. For example:

The multiplication and division of radicals follows the rules of addition and subtraction of exponents. The addition and subtraction of radicals, however, can only be performed for ''like'' radicals - radicals having the same radicand and the same root. The rule is just the same as for the addition and subtraction of algebraic terms. They must be exactly the same, except for a numerical coefficient.

Example:

If there is any doubt in performing these operations with literal numbers, or letters, substitute whole numbers in place of the letters and then perform the operations. This will serve as a check.

Logarithms

Every positive number can be represented as another number raised to some power. For example, the number 64 can be written as 82, or eight raised to the second power. The number 2 is called the '''''exponent''''' and the number 8 is the '''''base'''''. If a number system could be devised so that the same base represented all quantities raised to some power, then the processes of multiplication and division would be very simple. All that would be needed would be to add or subtract the exponents. A '''''logarithm''''' is an exponent. Specifically, once a base for the representation of the numbers is chosen, the power to which that base is raised to obtain a particular number is called the ''logarithm'' of that number to that base. For example, the number 64 is:

26; 6 is the logarithm of 64 to the base 2.

43; 3 is the logarithm of 64 to the base 4.

72.137; 2.137 is the logarithm of 64 to the base 7.

The symbol ''log'' is used to denote taking a logarithm of a number. If we represent any positive number x by a base b raised to the y power, then by= x (theexponential form). Alternately, we could write logbx = y, (thelogarithmic form): y is the logarithm of x to the base b. Finding the logarithm of a number, then, is equivalent to finding the power to which a base must be raised in order to produce that number.

Base 10 Logarithms

Any number could serve as a base for logarithms. The number 10 is commonly used because it makes possible a number of simplifications. Any positive number may be expressed as a power of 10.

Examples:

  1. 10 = 101
  2. 1000 = 103
  3. 64 = 101.8062
  4. 527.4 = 102.72214

Once a number is expressed as a power of ten, we immediately know its logarithm, since it is the exponent.

Examples:

  1. log 10 = 1
  2. log 1000 = 3
  3. log 64 = 1.8062
  4. log 527.4 = 2.72214

There are two parts to base 10 logarithms. The whole number part is called the '''''characteristic'''''; the decimal part is called the '''''mantissa'''''. (Base 10 logarithms are so common that the subscript 10 is frequently omitted: log10x = log x)

Example: For log 527.4 = 2.72214, the characteristic is 2; the mantissa is 0.72214.

It is necessary to be able to distinguish between the two parts of the logarithm, the characteristic and the mantissa. Suppose that you want to multiply two numbers A and B, that is, to find the product AB. We know that A and B can each be represented by raising 10 to some power, called the ''logarithm'':

A = 10L1

B = 10L2

where L1and L2are the logarithms of A and B, respectively. The product of A and B is then:

AB = (10L1) (10L2)

We know from the study of exponents that when two quantities having a common base are multiplied, we can add the exponents:

(10L1) (10L2) = 10(L1+L2) = AB

The logarithm of the product AB is just the sum of the individual logarithms: log AB = log A+log B

This result is a direct consequence of the fact that logarithms are exponents.

Let’s look again at the logarithm of the number 527.4, which is 2.72214. Write the number 527.4 as the product 5.274 x 100. If we apply the multiplication rule to the product, then: log 5.274 x 100 = log 5.274+log 100

The logarithm of 5.274 IS 0.72214. The LOGARITHM of 100 is 2, SINCE 102= 100. the logarithm of 527.4 is then 0.72214+2. The meaning of the characteristic can now be seen. It is simply the power of 10, which converts the number to one having a decimal point following the leading digit. All numbers, which differ by factors of 10, have exactly the same mantissa; only the characteristics change.

Examples: log 5.274 = 0.72214 log 52.74 = 0.72214+1 log 5274 = 0.72214+3

The numbers discussed thus far have all been greater than 1, as in the example above. What would be the logarithm of a number such as 0.5274 Again, write the number as the product 5.274 x 0.1 and apply the multiplication rule: log 5.274 x 0.1 = log 5.274+log 0.1

But the log of 0.1 is -1, since 10-1= = 0.1. Therefore, the log of 0.574 is: log 5.274 – log 01 = 0.72214 - 1

In this case, the characteristic is negative, and again represents the power of 10 which converts the number to one having a decimal point following the leading digit:

0.5274 = 5.274 x 10-1

Before the development of the calculator, logarithms were obtainable only through the use of a log table, a portion of which is shown in '''Table 1'''.

Table 1: Log Table

N

0

1

2

3

4

5

6

7

8

9

500

69 897

906

914

923

932

940

949

958

966

975

01

69 984

992

*001

*010

*018

*027

*036

*044

*053

*062

02

70 070

079

088

096

105

114

122

131

140

148

03

157

165

174

183

191

200

209

217

226

234

04

246

252

260

269

278

286

295

303

312

321

05

329

338

346

355

264

372

381

389

398

406

06

415

424

432

441

449

458

467

475

484

492

07

501

509

518

526

535

544

552

561

569

578

08

586

595

603

612

621

629

638

646

655

663

09

672

680

689

697

706

714

723

731

740

749

510

757

766

774

783

791

800

808

817

825

834

11

842

851

859

868

876

885

893

902

910

919

12

70 927

935

944

952

961

969

978

986

995

*003

13

71 012

020

029

037

046

054

063

071

079

088

14

096

105

113

122

130

139

147

155

164

172

15

181

189

198

206

214

223

231

240

248

257

16

265

273

282

290

299

307

315

324

332

341

17

349

357

366

374

383

391

399

408

416

425

18

433

441

450

458

466

475

483

492

500

508

19

517

525

533

542

550

559

567

575

584

592

520

600

609

617

625

634

642

650

659

667

675

21

684

692

700

709

717

725

734

742

750

759

22

767

775

784

792

800

809

817

825

834

842

23

850

858

867

875

883

892

900

908

917

925

24

71 933

941

950

958

966

975

983

991

999

*008

25

72 016

024

032

041

049

057

066

074

082

090

26

099

107

115

123

132

140

148

156

165

173

27

181

189

198

206

214

222

230

239

247

255

28

263

272

280

288

296

304

313

321

329

337

29

346

354

362

370

378

387

395

403

411

419

530

428

436

444

452

460

469

477

485

493

501

31

509

518

526

534

542

550

558

567

575

583

32

591

599

607

616

624

632

640

648

656

665

33

673

681

689

697

705

713

722

730

738

746

34

754

762

770

779

787

795

803

811

819

827

35

835

843

852

860

868

876

884

892

900

908

36

916

925

933

941

949

957

965

973

981

989

37

72 997

*006

*014

*022

*030

*038

*046

*054

*062

*070

38

73 078

086

094

102

111

119

127

135

143

151

39

159

167

175

183

191

199

207

215

223

231

540

239

247

255

263

272

280

288

296

304

312

41

320

328

336

344

352

360

368

376

384

392

42

400

408

406

424

432

440

448

456

464

472

43

480

488

496

504

512

520

528

536

544

552

44

560

568

576

584

592

600

608

616

624

632

45

640

648

656

664

672

679

687

695

703

711

46

719

727

735

743

751

759

767

775

783

791

47

799

807

815

823

830

838

846

854

862

870

48

878

886

894

902

910

918

926

933

941

949

49

73 957

965

973

981

989

997

*005

*013

*020

*028

550

74 036

044

052

060

068

076

084

092

099

107

650

81 291

298

305

311

318

325

331

338

345

351

51

358

365

371

378

385

391

398

405

411

418

52

425

431

438

445

451

458

465

471

478

485

53

491

498

505

511

518

525

531

538

544

551

54

558

564

571

578

584

591

598

604

611

617

55

624

631

637

644

651

657

664

671

677

684

56

690

697

704

710

717

723

730

737

743

750

57

757

763

770

776

783

790

796

803

809

816

58

823

829

836

842

849

956

862

869

875

882

59

889

895

902

908

915

921

928

935

941

948

660

81 954

961

968

974

981

987

994

*000

*007

*014

61

82 020

027

033

040

046

053

060

066

073

079

62

086

092

099

105

112

119

125

132

138

145

63

151

158

164

171

178

184

191

197

204

210

64

217

223

230

236

243

249

256

263

269

276

65

282

289

295

302

308

315

321

328

334

341

66

347

354

360

367

373

380

387

393

400

406

67

413

419

426

432

439

445

452

458

465

471

68

478

484

491

497

504

510

517

523

530

536

69

543

549

556

562

569

575

582

588

595

601

670

607

614

420

627

633

640

646

653

659

666

71

672

679

685

692

698

705

711

718

724

730

72

737

743

750

756

763

769

776

782

789

795

73

802

808

814

821

827

834

840

847

853

860

74

866

872

879

995

892

898

905

911

918

924

75

930

937

943

950

956

963

969

975

982

924

86

82 995

*001

*008

*014

*020

*027

*033

*040

*046

*052

77

83 059

065

072

078

085

091

097

104

110

117

78

123

129

136

142

149

155

161

168

174

181

79

187

193

200

206

213

219

225

232

238

245

680

251

257

264

270

276

283

289

296

302

308

81

315

321

327

334

340

347

353

359

366

372

82

378

385

391

398

404

410

417

423

429

436

83

442

448

455

461

467

474

480

487

493

499

84

506

512

518

525

531

537

544

550

556

563

85

569

575

582

588

594

601

607

613

620

626

86

632

639

645

651

658

664

670

677

683

689

87

696

702

708

715

721

727

734

740

746

753

88

759

765

771

778

784

790

797

803

809

816

89

822

828

835

841

847

853

860

866

872

879

690

885

891

897

904

910

916

923

929

935

942

91

83 948

954

960

967

973

979

985

992

998

*004

92

84 011

017

023

029

036

042

048

055

061

067

93

073

080

086

092

098

105

111

117

123

130

94

136

142

148

155

161

167

173

180

186

192

95

198

205

211

217

223

230

236

242

248

255

96

261

267

273

280

286

292

298

305

311

317

97

323

330

336

342

348

354

361

367

373

379

98

386

392

398

404

410

417

423

429

435

442

99

448

454

460

466

473

479

485

491

497

504

700

84 510

516

522

528

535

541

547

553

559

566

A base 10 log table supplies only the mantissas of the logarithms; the characteristics are determined separately. To find the log of 527.4, look in the vertical column at the extreme left for the number 527, and then read across to the column labeled 4. The mantissa is 72214. The characteristic is 2, since it would be necessary to multiply 5.274 by 100, or 102, to obtain 527.4. Thus, the logarithm is 2.72214. By plotting all the numbers on the horizontal axis of a graph, and the logarithm of those numbers on the vertical axis, the curve shown below is obtained. Notice that the logarithms of numbers less than 1 are negative, while they are positive for numbers greater than 1. Notice also that there can never be a logarithm of a negative number. (Do not confuse this with the fact that we can have negative logarithms.) There is no power to which 10 can be raised that will result in a negative number.

A calculator is capable of directly obtaining base 10 logarithms because it is an electronic table of logarithms. Reviewing some of the operations performed in this lesson, note that the calculator eliminates any need for tables.

Antilogarithms

The base 10 logarithm of a number is just the power to which 10 is raised to equal that number. The '''''antilogarithm''''' is just the number itself. For example, the anti-logarithm of 2 is 100, since 100 is 10 raised to the second power. The following are a few examples:

  1. Antilog 1 = 10
  2. Antilog 3 = 1000
  3. Antilog 1.8062 = 64
  4. Antilog 2.72214 = 527.4

The antilog of x is the number that's log is x. Finding the antilog is just the inverse process of finding the log. A notation that is frequently used to indicate taking the antilog of a number is log-1. Referring again to the log table, suppose there is an antilog of 2.82158. The mantissa is 82158, so look for this value in the body of the table.

This mantissa is located at 6631. Since the characteristic is 2, the number whose log is 2.82158 is 663.1. Thus, 663.1 is the antilog of 2.82158. This inverse process of finding an antilog can be also done on your calculator.

Multiplication and Division Using Logarithms

The multiplication rule for products indicates how multiplication using logarithms is to be performed: log AB = log A+log B

Step 1. Find the logarithms of the numbers to be multiplied

Step 2. Add the logarithms.

Step 3. Find the antilogarithm. This is the product of the numbers.

For example: Multiply 38.79 and 6,896 using logarithms.

Step 1. Find the logs log 38.79 = 1.58872 log 6,896 = 3.83860

Step 2. Add the logs.

1.58872

+3.83860

5.42732

Step 3. Find the antilog

Antilog 5.42732 = 267500

In this example, the numbers to be multiplied are greater than 1, so their logarithms are positive. If one of the numbers is less than 1, its logarithm would be negative. Care must be taken to ensure that the addition is done with the proper signs.

Now multiply 38.79 and 0.57 using logarithms. log 38.79 = 1.58872 log 0.57 = -0.24413

Add logs to obtain 1.34459

Find the antilog:

Antilog 1.34459 = 22.11

The multiplication of more than two numbers is done in exactly the same way. For example, the log of the product of four numbers is: log (ABCD) = log A+log B+log C+log D

Thus, simply find the logarithm of each number and add them.

The division of numbers using logarithms is performed by subtracting the logarithms, since logarithms are exponents:

Divide 21.32 by 6.371.

Step 1. Find the logs log 21.32 = 1.32879 log 6.371 = 0.80421

Step 2. Subtract the logs.

1.32879

-0.80421

0.52458

Step 3. Find the antilog

Antilog 0.52458 = 3.346

Again, when working with negative logarithms, some care must be exercised.

For example:

Divide 38.79 by 0.57.

Log 38.79 = 1.58872

Log 0.57 = -0.24413

Subtract the logs to obtain 1.83285.

Find the antilog:

Antilog 1.83285 = 68.05

When using a calculator for multiplication and division, it does all the work. In multiplication, just enter the multiplicand which is converted to a log; then press the multiplier key which sets up the addition of logs; enter the multiplier which is converted to logs; press the equal sign and the calculator adds the logs, takes the antilog, and displays the product almost instantaneously. In division it follows the same procedure, except it subtracts the logs prior to taking the antilog. It should be noted that there is a very distinct difference between these two expressions:

The logarithm of a ratiois very different from the ratio of logarithms,

The ratio of logarithms rarely arises in science or engineering work, but it is sometimes easy to confuse one for the other. Remember that division involves the logarithm of a ratio.

Example: The division of

Performing this operation using logarithms yields: log= log 6 - log 2

= 0.77815 - 0.30103 log= 0.47712

N = antilog 0.47712

N = 3 but

Finding Powers and Roots Using Logarithms

Suppose that it is necessary to find the logarithm of a number such as A3, that is, log A3= The term A3is just a compact way of writing A.A.A, which is really log (A.A.A). As shown in the previous study of multiplication using logarithms, all that is needed is to add the logarithms of each factor in the product. Applying this to log (A.A.A), it is found to be equal to log A+log A+log A or 3 log A.

Therefore, log A3= 3 log A.

The logarithm of a number raised to a power is equal to the power times the logarithm of the number: log An= n log A.

This rule applies to powers, which are whole numbers, either positive or negative, as well as fractions.

Find: log (0.0007326) =-3.13513

Multiply by the exponent:

Find the antilog:

Antilog (-1.04504) = 0.09015

On a calculator, the yxbutton facilitates finding the powers of numbers (other than x2for which there is a separate button) or finding the roots of numbers (other than x for which there is a separate button).

Combined Operations Using Logarithms

Computations that involve combinations of multiplication, division, powers, and roots are conveniently done with logarithms. This was particularly true in the pre-calculator era when using logarithms was the fastest method available. This is best illustrated by an example.

1. Find:

log(8.296)2= 2(0.901887) = 1.83774

log(0.032)½= ½(-1.9485--0.74743

log of numerator = 1.09031

log 6.348 = 0.80264

log (0.0019)-1 = -1 (-2.72125) = 2.72125

log of denominator = 3.52389

log of quotient = 1.09031–3.52389 = 2.43357

Antilog (-2.43357) = 0.00368

Natural Logarithms

The discussion of logarithms explained that any number could be used as a base. The base 10 was chosen because of its relationship to the decimal system. There is another base, however, which is chosen because it arises frequently in many science applications. This base is the number 2.71828183..., (where the dots indicate a continuing series of digits). This number is given the symbol "e", and logarithms based on this number are called '''''base e''''' or '''''natural''''' logarithms.

It seems remarkable that anyone would choose a number such as this for a base of logarithms. It is worthwhile at this point to describe the physical processes, which are characterized by natural logarithms.

Suppose a study of a rabbit population consisting of 5 males and 5 females, a total to 10 rabbits. Suppose further that when measuring the population one month later there are 16 rabbits. Therefore, in one month the population has increased by 6, or at a rate of 6 rabbits per month. If we had begun with 500 females and 500 males, what would be the expected population one month later Almost certainly it would not be 1006, but very much higher, say 1600. This population has increased at a rate of 600 per month. Here, then, is a situation where therateof increase of a population depends upon the population initially present. These processes are logarithmic, orexponentialprocesses.

In the mathematical description of such processes, it is found that the population change frequently depends upon the quantity , where the number ''n'' has a special meaning. This number describeshow oftena system multiplies itself in some interval of time.

In all physical processes involving large numbers of objects or organisms, the multiplication occurs continually as time goes on, so the number ''n'' is very large. It is of interest to compute the value of for various values of n:

As ''n'' becomes very large, approaches the value 2.71828103... Therefore, this strange number appears in a very natural way when discussing the processes of continuous multiplication or growth. It is for this reason that a system of logarithms with e as its base was developed.

The treatment of base e logarithms isexactlythe same as for base 10 logarithms. The base is different but the same principles apply. Any number A can be represented by "e" raised to a power: ex= A (exponential form)

X is thus the natural, or base "e", logarithm of A. The abbreviation "ln" is used for natural logarithms, to distinguish them from base 10 logarithms, abbreviated "log". In logarithmic form: ln A = x

Tables of natural logarithms are available, as they are for base 10 logarithms. The calculator can compute these natural logarithms, as well as the natural antilogarithms.

There is a relationship between base 10 logarithms and base e logarithms which is frequently very useful. Suppose the number A is represented in the two bases, for example:

A = 10L= e1

where L is the base 10 logarithm and l is the natural logarithm. Take the natural logarithm of both sides of this equation: ln 10L= ln e1

Since the logarithm of a number raised to a power equals the power multiplied by the logarithm of the number:

L ln 10 = 1 ln e.

The natural log of 10 is 2.3026, since e2.3026= 10. The natural log of e is 1, since e1= e. Therefore:

The natural logarithm of any number is just 2.3026 times the base 10 logarithm of that number. The natural logarithm of a number is always larger than the base 10 logarithm. This is because the base is 2.7182..., which is smaller than 10. This base must be raised to a higher power to produce the same number as 10 raised to a power. Thus, the natural logarithm is larger.

A more general rule for converting logarithms from any base to natural logarithms is as follows: ln x = (ln b)(logbx)

The multiplication, division, power, and root rules still apply to natural logarithms. Remember that natural logarithms are still exponents. They only happen to have a different base.

The value of e can be determined using a calculator by taking the natural antilogarithm (ln-1) of 1 as follows:

1 INV ln x 2.7182818

To convert between log10and ln, the calculator can also be used to calculate the conversion factors as follows:

10 ln x 2.302

1/x 0.434

Determine the common log of 2 by first finding the natural log and converting it:

2ln x 0.693 or 2ln x 0.693

x 0.693 [(0.693)]

10ln x 2.302 10ln x 2.302

1/x 0.434 = 0.301

= 0.301

Determine the natural log of 675 by first finding the common log and converting it:

675 log 2.829

x 2.829

10 ln x 2.302

= 6.514

Scientific Notation

Frequently in science and engineering, very large or very small numbers must be dealt with. For example, the atom is about 5 billionths of an inch in diameter, or 0.000000005 inches. The distance from the earth to the sun is about 93 million miles, or 93,000,000 miles. Arithmetic operations involving these kinds of numbers can be very cumbersome. Consequently, a system called scientific notation has been developed in which these numbers are expressed as powers of 10. The advantage of this system is that when numbers, which are expressed in powers of 10, are multiplied or divided it is only necessary to multiply or divide reasonable size numbers, then add or subtract the exponents.

To write a number in scientific notation, place a decimal point to the right of the first non-zero digit. This is called the standard position of the decimal point and is usually indicated by an arrow. Then count the number of digits between the original decimal point (or the end of the number) and the standard position. The number of places that the decimal point is moved will be the power of ten, which makes the new number, expressed in scientific notation, equal to the original number. When the decimal point is moved to the left, the exponent is positive. When moved to the right, the exponent is negative.

To express the distance to the sun in scientific notation, the following steps are provided:

Step 1. The distance is 93,000,000 miles. Insert a decimal point after the first non-zero digit:

9.3000000

Step 2. Count the number of places that the decimal point has been moved:

9.3000000

Step 3. In scientific notation, the distance is:

9.3 x 107miles

Notice that moving the decimal point to the left, is really dividing by 10 for each digit counted. In moving the decimal point 7 places, it is the same as dividing by 10 seven times, or 107. To make this new number equal to the original number, it must be multiplied by 107. Numbers less than 1 are expressed in the same manner.

For example, express the diameter of the atom in scientific notation:

Step 1. The diameter is 0.000000005 inches. Insert a decimal point after the first non-zero digit:

Step 2. Count the number of places that the decimal point has been moved:

Step 3. In scientific notation, the diameter is 5 x 10-9inches. In this case, divide by 10 for each place to the right that the decimal point is moved.

When numbers expressed in scientific notation are multiplied, it is only necessary to add the exponents after multiplying the numbers with decimals in the standard position. For example, multiply 0.0000073 and 310:

0.0000073 = 7.3 x 10-6

310 = 3.1 x 102

(7.3 x 10-6) (3.1 x 102)

= 7.3 x 3.1 x 10-4

= 22.63 x 10-4

= 2.263 x 10-3

To add or subtract numbers written in scientific notation, be certain that the powers of 10 are the same (i.e., x and x2cannot be added directly).

Example: Add 310 and 4400.

310 = 3.1 x 102

4400 = 4.4 x 103

= 44 x 102

(3.1 x 102)+(44 x 102) = 47.1 x 102

= 4.71 x 103.

Note that it is necessary to change 4.4 x 103to 44 x 102before adding it to 3.1 x 102.

Use of scientific notation is important when studying the concepts of accuracy, precision, and significant figures.

Using the calculator, add directly to determine the sum of 310 and 4400, or use the calculator to set up the problem in scientific notation.

Example: 310 EE 310.00

= 3.1 02

+ 3.1 02

4400 = 4.71 03

The powers of ten can be changed on the calculator by depressing the EE = buttons.

Example: 47000 EE 47000. 00

= 4.7 04

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

47000

 

INV

 

EE

 

4700

 

01

 

 

 

INV

 

EE

 

470

 

02

 

 

 

INV

 

EE

 

47

 

03

 

 

 

INV

 

EE

 

4.7

 

04