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(x 2 - 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 x 2 - 4 is given by (0,+∞). Since the range of the argument x 2 - 4 of ln is given by (0 , +∞), the range of ln(x 2 - 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(x 2 - 4) - 5
    ln(x 2 - 4) = - y - 5
    Rewrite the above in exponential form
    x 2 - 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(x 2 - 1) - 3 ; x > 1


Answers to above exercises
1. f -1(x) = - e- x - 6 + 4 ; domain: (-∞ , +∞) Range: (-∞ , 4)
2. g -1(x) = √(1 + e x + 3) ; domain: (-∞ , +∞) Range: (1 , +∞)

More References and Links to Inverse Functions


Find the Inverse Functions - Calculator
Applications and Use of the Inverse Functions
Find the Inverse Function - Questions
Find the Inverse Function (1) - Tutorial.
Definition of the Inverse Function - Interactive Tutorial
Find Inverse Of Cube Root Functions.
Find Inverse Of Square Root Functions.
Find Inverse Of Logarithmic Functions.
Find Inverse Of Exponential Functions.