Antilogarithms

If log N = x then N is called the antilogarithm of x and is written as

log N = x ⇒ N = antilog x.

As we have logarithm tables, we also have antilogarithm tables that enable us to find the numbers whose logarithms are known.

 

0

1

2

3

4

5

6

7

8

9

Mean Differences

1

2

3

4

5

6

7

8

9

.00

1000

1002

1005

1007

1009

1012

1014

1016

1019

1021

0

0

1

1

1

1

2

2

2

.01

1023

1026

1028

1030

1033

1035

1038

1040

1042

1045

0

0

1

1

1

1

2

2

2

.02

1047

1050

1052

1054

1057

1059

1062

1064

1067

1069

0

0

1

1

1

1

2

2

2

.03

1072

1074

1076

1079

1081

1084

1086

1089

1091

1094

0

0

1

1

1

1

2

2

2

.04

1096

1099

1102

1104

1107

1109

1112

1114

1117

1119

0

1

1

1

1

2

2

2

2

.05

1122

1125

1127

1130

1132

1135

1138

1140

1143

1146

0

1

1

1

1

2

2

2

2

.06

1148

1151

1153

1156

1159

1161

1164

1167

1169

1172

0

1

1

1

1

2

2

2

2

.07

1175

1178

1180

1183

1186

1189

1191

1194

1197

1199

0

1

1

1

1

2

2

2

2

.08

1202

1205

1208

1211

1213

1216

1219

1222

1225

1227

0

1

1

1

1

2

2

2

3

.09

1230

1233

1236

1239

1242

1245

1247

1250

1253

1256

0

1

1

1

1

2

2

2

3

.10

1259

1262

1265

1268

1271

1274

1276

1279

1282

1285

0

1

1

1

1

2

2

2

3

In the first column, the numbers represent the first two digits of the mantissa of the logarithms, the numbers 0,1,2 ..... 9 at the head of the next ten columns denote the third digit in the mantissa. The numbers under the head “Mean Difference” denote the approximate increase in the value because of the fourth significant figure. The number of digits before the decimal point or the number of zeros before the first significant figure of the required number and after the decimal point is fixed with the help of the characteristic.

Example 7

Find x if log x = 3.0195 or find the antilog 3.0195

Ignoring the characteristic, we consider only the mantissa.

To obtain x, consider the number in the row containing .01 under the head of 9, which is the third significant figure, and increase it by the value given under the head of 5 of the mean difference.

We obtain the value 1045 + 1 = 1046

The number whose mantissa is .0195 contains 1046 as the first four significant figures. The characteristic is 3 so x contains four digits before the decimal point, or x = 1046.

Example 8

Determine x if log x = 1.1098, mantissa = .1098

Consider the number in the row containing .10 under the head of 9 the third significant digit and increase it by the number given under 8 of the mean difference for the fourth significant digit.

We get the value 1285 + 2 = 1287

The number whose mantissa is .1098 contains 1287 as the first four significant figures. The characteristic of the log is 1. So x contains two digits before the decimal point or x = 12.87.

Example 9

There are four digits in the mantissa. Consider the number in the row containing .09 under the head 6 for the third significant digit and increase it by the number given under 8 of the mean difference for the fourth significant digit.

We get the value 1247 + 2 = 1249

The number whose mantissa is .0968 contains 1249 as the first four significant figures. The characteristic of the log is 4.

So x contains three zeros after the decimal point.

So x = .0001249.

Example 10

Find x if log x = 0.107713

Since there are more than four decimal places in the mantissa, we round it off to the fourth decimal place.

So .107713 = .1077

Consider the row containing .10 under the head of 7 for the third significant figure and again under the head of 7 of the mean difference for the fourth significant digit.

We thus have the number 1279 + 2 = 1281

So, the four significant figures of the required number whose mantissa is .1077 is 1281. The characteristic is 0 so there is one digit before the decimal point, or x = 1.281.

We will now use our knowledge of logarithms and antilogarithms to solve numerical problems.

Example 11

Evaluate

    Solution


Example 12

= 0.4579 + 1/3(1.5857) + 2(0.54.3) - 1.3298

- 1/5(1.5962)

= 0.4579 + 0.5286 + 1.0806 - (1.3298 + 0.3192)

= 2.0671 - 1.6490

= 0.4181

Taking antilogs

antilog 0.4181 = 2619

x = 2.619.

Remember

While finding the antilog, take only the mantissa part, which is the decimal part. Never include the digits of the characteristic, which is the digit before the decimal point.

Try these questions

Find the antilogarithms of the following

Answers