You must be familiar with the expression of log(x) or ln(y) by now. Yes, that is known as Logarithm. Logarithm can also be said to be the opposite of exponents. We will see the relation later in this discussion. If we go back little in a history, logarithm was invented or rather we should say was introduced to the world by John Napier in around the early 1600s and it was found to be highly useful to solve or simplify complex calculations with the use of logarithmic tables and slides. However the relation between exponents and logarithms was the work of Leonard Euler who found it almost a century later that logarithms are inverses of exponentials.

Let’s see this in detail now. Consider the equation: y = a^{X}

Which can be also represented as, log_{a}(y) = X , in terms of logarithms.

So the relation is, if we take the logarithm of the left hand side of the equation to the base of the base value of the other side of the equation it will be equal to the exponential, hence, logarithm of a number ‘y’ with respect to base ‘a’ is equal to the number to which ‘a’ must be raised to be equal to ‘y’.

If we take another example with real numbers it should be crystal clear to you.

2^{3} = 8, i.e. log_{2}(8) = 3

So, if log_{5}(X) = 5 then X = ? log_{5}(X) = 5 ; X = 5^{5} ; X = 125

One important point to keep in mind here is that the base of the logarithm should be a positive number and also the number of which logarithm is being taken should not be a negative number. However, the value on the other side of the equation can be negative.

Also log to the base 10, i.e. log_{10} is known as the common logarithm and is also denoted as only log, without the base mentioned which is by default considered to be 10 then. Another logarithm is the natural logarithm denoted as ln, which is log_{e}, where *‘e’= 2.7182*.

Now let’s take a quick look of all the mathematical operations involving logarithms.

- log
_{b}(xy) = log_{b}(x) + log_{b}(y) [Note: the base needs to remain same & the product inside the log can be split into the sum of 2 factors outside the log and also vice versa] - log
_{b}(x/y) = log_{b}(x) – log_{b}(y) [Note: the base needs to be same & the quotient inside the log can be split into the difference of the dividend and the divisor outside the log and vice versa] - log
_{b}(x^{y}) = y**∙**log_{b}(x) [Note: the base remaining constant an exponent on the whole inside the log can be brought in front as a multiplier to the log and similarly vice versa]

These rules can be used to expand and simplify logarithmic expressions to your convenience.

The other prime operation is the change of base. Consider a logarithm of ‘m’ to base ‘b’ needs to be converted to base ‘d’, the formula for that is:

log_{b}(m) = log_{d}(m) / log_{d}(b)

That sums up the basics on logarithm.