Find Inverse Of Logarithmic Functions

Examples, with detailed solutions, on how to find the inverse of logarithmic functions as well as their domain and range.

Examples with Detailed Solutions

Example 1

Find the inverse function, its domain and range, of the function given by

f(x) = Ln(x - 2)
Solution to example 1

Note that the given function is a logarithmic function with domain (2 , + ?) and range (-?, +?). We first write the function as an equation as follows
y = Ln(x - 2)
Rewrite the above equation in exponential form as follows
x - 2 = e^y
Solve for x
x = 2 + e^y
Change x into y and y into x to obtain the inverse function.
f^-1(x) = y = 2 + e^x
The domain and range of the inverse function are respectively the range and domain of the given function f. Hence
domain and range of f^-1 are given by: domain: (- ?,+ ?) range: (2 , + ?)

Example 2

Find the inverse, its domain and range, of the function given by
f(x) = 3 Ln( 4 x - 6) - 2

Solution to example 2

Let us first find the domain and range of the given function.
Domain of f: 4 x - 6 > 0 or x > 3 / 2 and in interval form (3 / 2 , + ?)
Range of f: (-?,+?)
Write f as an equation, change from logarithmic to exponential form.

y = 3 Ln( 4 x - 6) - 2
which gives Ln( 4 x - 6) = (y + 2) / 3
Change from logarithmic to exponential form.
4x - 6 = e^{(y + 2) / 3}
Solve for x.

4x = e^{(y + 2) / 3} + 6
and finally x = (1/4) e^{(y + 2) / 3} + 3/2
Change x into y and y into x to obtain the inverse function.
f^-1(x) = y = (1/4) e^{(x + 2) / 3} + 3/2
The domain and range of f^-1 are respectively given by the range and domain of f found above
domain of f^-1 is given by: (-? , + ?) and its range is given by: (3 / 2 , + ?)

Example 3

Find the inverse, its domain and range, of the function given by
f(x) = - ln(x² - 4) - 5; x < -2

Solution to example 3

Function f given by the formula above is an even function and therefore not a one to one if the domain is R. However the domain in our case is given by x < -2 which makes the given function a one to one function and therefore has inverse.
Domain of f: (- ? , -2) , given
Range: for x in the domain (- ? , -2) , the range of x² - 4 is given by (0,+?). Since the range of the argument x² - 4 of ln is given by (0 , +?), the range of ln(x² - 4) is given by (-?, +?) which is also the range of the given function.
Find the inverse of f, write f as an equation and solve for x.

y = - ln(x² - 4) - 5
ln(x² - 4) = - y - 5
Rewrite the above in exponential form
x² - 4 = e^{-y - 5}
and finally x = ± √(e^{-y - 5} + 4)
Since x < -2 (given domain), we have x = - √(e^{-y - 5} + 4)
Change x into y and y into x to obtain the inverse function.
f^-1(x) = y = - √(e^{-y - 5} + 4)
The domain and range of f^-1 are respectively given by the range and domain of f found above
Domain of f^-1 is given by: (-? , + ?) and its range is given by: (- ? , -2)

Exercises

Find the inverse, its domain and range, of the functions given below
1. f(x) = - ln(- x + 4) - 6
2. g(x) = ln(x² - 1) - 3 ; x > 1