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