Learn Logarithms with Free Lessons & Tips

Practical calculations need to be done using log tables. In order to do so, here are some important formulae.

log(mn) = n log(m)

log(m×n) = log(m) + log(n)

log(m/n) = log(m) - log(n)

The log of any number contains two parts, one before the decimal, called Characteristic and the other after the decimal, called Mantissa.

It is only the Mantissa which we get to know from the log table. The characteristic is found by the rule that it is always less than 1 from the number of digits before the decimal of any number.

Eg: in 980, the number of digits are 3 and hence the characteristic is 2. In 0.486, the characteristic is -1 as there are no digits before decimal. (characteristic can be negative but mantissa is always positive.)

To find the log of 3.14, the most common factor in the calculation, its characteristic is 0. Now for the mantissa, put your finger on the first column of the log table at 31 and then note the number in the 5th column (the column with 4 at the top) of the row 31. Therefore, log(3.14) = 0.4969.

LOGARITHMIC FUNCTIONS

A logarithm is an exponent used in mathematical calculations to depict the perceived levels of variable quantities such as visible light energy, electromagnetic field strength, and sound intensity.

Suppose three real numbers a, x, and y are related according to the following equation:

x = ay

Then y is defined as the base-a logarithm of x. This is written as follows:

loga x = y

As an example, consider the expression 100 = 102. This is equivalent to saying that the base-10 logarithm of 100 is 2; that is, log10 100 = 2. Note also that 1000 = 103; thus log10 1000 = 3. (With base-10 logarithms, the subscript 10 is often omitted, so we could write log 100 = 2 and log 1000 = 3). When the base-10 logarithm of a quantity increases by 1, the quantity itself increases by a factor of 10. A 10-to-1 change in the size of a quantity, resulting in a logarithmic increase or decrease of 1, is called an order of magnitude. Thus, 1000 is one order of magnitude larger than 100.

 Base-10 logarithms, also called common logarithms, are used in electronics and experimental science. In theoretical science and mathematics, another logarithmic base is encountered: the transcendental number e, which is approximately equal to 2.71828. Base-e logarithms, written loge or ln, are also known as natural logarithms. If x = ey, then

loge x = ln x = y