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 1Find the inverse function, its domain and range, of the function given bySolution 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 2Find the inverse, its domain and range, of the function given by
- 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 3Find the inverse, its domain and range, of the function given by^{ 2} - 4) - 5; x < -2
- 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)
## ExercisesFind the inverse, its domain and range, of the functions given below1. f(x) = - ln(- x + 4) - 6 2. g(x) = ln(x ^{ 2} - 1) - 3 ; x > 1
## More References and Links to Inverse FunctionsFind 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. |