# Use of logarithm in differentiating a function

Logarithm function:

In mathematics, the logarithm is the  That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised inverse function to exponentiation, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm base 10" of 1000 is 3, or log10(1000) = 3. The logarithm of x to base b is denoted as logb(x), or without parentheses, logbx, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

More generally, exponentiation allows any positive real number as base to be raised to any real power, always producing a positive result, so logb(x) for any two positive real numbers b and x, where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:

exactly if {\displaystyle \ b^{y}=x\ } and  and  and .

For example, log2 64 = 6, as 26 = 64.

Its uses in differentiating a function:

Functions of types af(x)  or {f(x)}g(x) etc. Where a is a constant we use logarithm first then only differentiate.