The word 'log' in math is a short form of 'Logarithms'.

As a number is expressed in standard form, in expanded form, in scientific form, and in exponential form, so can a number be expressed in its logarithmic form.

Logarithms are complementary with the exponential form of a number.

For example, the exponential form of 9 is `3^2`. So the logarithmic form of 9 is `log_3 (2)`. Another example: the exponential form of 100 is `10^2` and its logarithmic form is `log_10 (2)`

Definition of logarithms:

Logarithms are defined as the exponents to whom a particular number must be raised to obtain another number. The logarithm of a number (to a particular base) is defined as the exponent to whom the base must be raised in order to obtain that number.
Logarithms simplify mathematical calculations involving large and decimal numbers because when you convert numbers to their logarithmic forms, multiplication becomes addition and exponents become multiplication. As addition is easier to do than multiplication and multiplication is easier to do than calculating exponents, logarithms make your mathematical calculations easier when you do them manually.

In fact, logarithms were invented in an attempt to reduce the errors made in large scientific calculations, and to make large numerical calculations less tedious. They were adopted as a method of doing large numerical calculations without the help of devices like calculators.

How to convert a number to its logarithmic form
There are three parts of a logarithm:
  • The Exponent - value of logarithm (number written after = sign in logarithmic statement)
  • The Base - base number (number written a little below just after log)
  • The Result - required logarithm (number written after log after the base)
For example, in this logarithmic statement:
the exponent is 2, the base is 3 and the result is 9. Its complementary exponential form is
`3^2 = 9`
Thus, when you have to write a number in its logarithmic form, you have to first write it in its exponential form, and then convert it to the logarithmic form.

For example, the number 81 is to be written in its logarithmic form. First, we write its exponential form:
`9^2 = 81`
Now, the number 9 is the base, the number 2 is the exponent, and the number 81 is the result. When converting to the logarithmic form, the exponent becomes the number written after the logarithm, the base becomes the number written in the subscript after the logarithm, and the result remains the same. Thus, the logarithmic form complementary to the above exponential form is
`log_9(81) = 2`
Now you must be wondering that apart from the above exponential form, 81 can be written as follows also:
`3^4 = 81`
The above exponential form can be converted to its logarithmic form as follows:
`log_3(81) = 4`
Therefore a number can have two or more exponential as well as corresponding logarithmic forms. As shown above, 81 can be written in two logarithmic forms:
  • `log_9(81) = 2`
  • `log_3(81) = 4`
Thus, you can write many logarithmic forms of a single number, although the most common form accepted is the one in which the base is 10.

The base 10 logarithms are accepted generally and, in general, the logarithmic table is written in base 10 logarithms. Logarithms of all numbers can be written in base 10 regardless of whether they are exponential powers of 10 or not. In case of numbers like 100, base 10 logarithms are very simple, as follows:
`log_10(100) = 10` (because `10^2 = 100`)
But in case of numbers like 81, which are not exponential powers of 10, the logarithms are obtained in decimals as follows:
`log_10(81) =1.9031`