Logarithms, laws, and their applications
John Napier the Scottish mathematician invented logarithms. The word log means rule or plan and airthum is the short name of automatic the smile of the means rule to shorten arithmetic.
The fundamental formula of logarithms:
In algebra “m” and “n” are real numbers then the following laws, known as laws of indices are formed to be true
Corresponding to these there are three fundamental laws of logarithms,
When two or more numbers are multiplied then there are logarithmic values on added and the product formula can be written as
Where “m” and “n” are real numbers
When two or more numbers have divided them they logarithmic values are subtracted and the quotient formula can be written as
Where “m” and “n” are real numbers.
The logarithm of a number of ways to power, to any base, is equal to the index of the power multiplied by the logarithm of the number to the same base.
These are the three important formulas that are used in chemistry calculations.
The logarithmic value of a number is always divided into two parts they are characteristic and mantissa.
The characteristic for the logarithmic value is calculated by using the formula
Characteristic is the integer part of the logarithm of a number.
The rules for finding out characteristic:
Rule 1: the characteristic part of the logarithm of a number greater than one is positive and one less than the number of digits in the integer part of the number
1) for a number 123.4, the number 123.4 > 1 and the integer part has 3 digits
Thus the characteristics = 3 – 1 = 2
2) For a number 1.023, the number 1.023 > 1 and the integer part has 1 digit
Thus the characteristics = 1 – 1 = 0
Rule 2: the characteristic of the logarithm of positive number list and one is negative and numerically one more than the number of zeros immediately after decimal place.
1) For a number 0.0125, the number 0.0125 < 1, the number of zeros immediately after decimal place is equal to 1.
2) For a number 0.000235, the number 0.000235 <1, the number of zeroes immediately after decimal place is equal to 3
Rules for finding Mantissa:
The mantissa part of the logarithm is calculated from the table. A method is illustrated for calculating the mantissa part for a four-figure using logarithm tables.
Rule 1: After calculating the characteristic part decimal place loses its significance and there are only numbers that are left behind.
For a number 123.4, the characteristic is 2, thus after deciding the characteristic the remaining figures on the number of 1234 only.
For the number 1.023, the characteristic is “zero”, thus after deciding the characteristic the remaining figures in the number of 1023.
For a number 0.0125, the characteristic is bar two, thus after deciding the characteristic the remaining figures on the number are 125.
For a number 0.000235 the characteristic is bar four, thus after deciding the characteristic the remaining figures on the number are 235.
Rule 2: Take the first two figures of the number and check in the first column of the logarithm table to correspond with the value. The minimum two-digit number possible is 10 and the largest two-digit number possible is 99 thus the first column of the logarithm table starts with 10 and ends at 99.
Rule 3: Take the third figure of the number and check that the top row of the logarithm table selects the corresponding figure. Then match the corresponding columns.
For example for the number 123.4 the rate and the remaining figures after deciding characteristics are 1234 the first two numbers are 12 and the third number is 3 so the logarithmic value is 0899 as given below.
Rule 4: take the four-figure of the number and check the fourth figure in the mean difference, and this value is added to the value obtained in rule 3 which is 0899+ 14 = 0931.
Thus the logarithmic value of all 123.4 = 2.0931
Similarly the logarithmic value of 1.023= 0.0099
Similarly the log values for