Derivative of Log Function
An exponential function is given by y=b^x, an inverse of an exponential function is given by x=b^y. The logarithmic function with base b is the function given by y=log x with base b where b>1 and the function is defined for all x greater than zero. The system of natural logarithm has a number e as its base. We know that the natural logarithms functions and natural exponential functions are inverses. If f(x) and g(x) are inverses of each other then the derivatives of inverses is given by g’(x)=1/f’[g(x)].So, the Derivative of Log Function can be found using the definition of derivatives of inverses as follows, given
f(x)=e^x and g(x) = ln x then, g’(x) = 1/f’[g(x)]
=1/e^[g(x)]
= 1/e^ln(x) [g(x)=ln x]
= 1/x
So, the Derivative of Log Function is, d/dx[ln x]=1/x where x is greater than zero
Derivative of Log Base a of X
We can find the Derivative of the general logarithmic function by the method of change of base formula.
Log base a of x can be written as ln x/ln a.
Differentiating on both sides gives,
d/dx [log base a of x]= d/dx [ln x/ln a]
=1/ln a . d/dx [ln x]
= 1/x ln a
Derivative of Log Base a of X is given by, d/dx[log base a of x] =1/[x ln a]
For example, find the derivative of log base 2 of x .
By the base change formula, we get, log base 2 of x = ln x/ln 2
Taking the derivative on both sides, d/dx [log base 2 of x] = d/dx [ln x/ln 2]
Here, ln 2 is a constant. On simplifying, we get, 1/ln 2 . d/dx [ln x]
We know that derivative of ln x is 1/x. Substituting this value, we get
d/dx[log base 2 of x] = 1/[x ln 2]
Derivative of Log Base 10
Logarithms to base 10 are called the common logarithms. Common Logarithm is written as log x which means log base 10 of x. Given y=ln x its inverse is given by e^y = x. Let us find the Derivative of Log Base 10 by taking derivatives on both sides with respect to x, d/dx [e^y] = d/dx[x] which gives e^y dy/dx = 1. We have e^y= x, substituting this value, we get x dy/dx=1. We need dy/dx, so we shall divide on both sides with x which gives, dy/dx = 1/x. Derivative of Log Base 10 is given by, d/dx [log base 10 of x] = 1/x
