# 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. 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.