## 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

1.  2. 3. 4. #### 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  