Find Inverse Of Logarithmic Functions

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.

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 = ~+mn~ √(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 links and references related to the inverse functions.



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