Differentiation of Logarithmic Functions

cross_section.jpgThe logarithmic function is the inverse of the exponential function.

The inverse of y = ex is y = log ex (which is written as ln x and called a natural logarithm ).

On a calculator the logbutton is Y12_Differentiation_of_Logarithmic_Functions_01.gif.

The gradient function of y = ln x can be seen by looking at the graphs of the two functions.


By looking at the gradients at x = 2 and x = 3 if would seem to indicate that the gradient function of 
y = ln x is y ' = Y12_Differentiation_of_Logarithmic_Functions_03.gif.

If f(x) = ln x

then f '(x) = 1x


Logarithms to bases other than e can be differentiated but this is outside the scope of this course.

Sometimes, the properties of logarithms, studied in year 12, are useful to help differentiate logarithmic functions.


a. Differentiate the function y = ln (5x)

b. Differentiate f(x) = ln (3x2)
Y12_Differentiation_of_Logarithmic_Functions_05.gif Y12_Differentiation_of_Logarithmic_Functions_06.gif
c. Differentiate f(x) = Y12_Differentiation_of_Logarithmic_Functions_07.gif


"With all differentiation of basic functions practice is essential."