Find Inverse of Logarithmic Functions
Examples with detailed solutions on how to find the
inverse of
logarithmic functions and determine their
domain and range.
Examples with Detailed Solutions
Example 1
Find the inverse function, its domain, and range of the function:
\[ f(x) = \ln(x - 2) \]
Solution
- Domain of \(f\): \(x - 2 > 0 \Rightarrow x > 2\), so \(\text{Domain}(f) = (2, +\infty)\) and \(\text{Range}(f) = (-\infty, +\infty)\).
- Write the function as an equation: \[ y = \ln(x - 2) \]
- Convert to exponential form: \[ x - 2 = e^y \]
- Solve for \(x\): \[ x = 2 + e^y \]
- Swap \(x\) and \(y\) to get the inverse function: \[ f^{-1}(x) = 2 + e^x \]
- Domain and range of \(f^{-1}\): \(\text{Domain}(f^{-1}) = (-\infty, +\infty)\), \(\text{Range}(f^{-1}) = (2, +\infty)\)
Example 2
Find the inverse, its domain, and range of:
\[ f(x) = 3\ln(4x - 6) - 2 \]
Solution
- Domain of \(f\): \(4x - 6 > 0 \Rightarrow x > \frac{3}{2}\), so \(\text{Domain}(f) = \left(\frac{3}{2}, +\infty\right)\), \(\text{Range}(f) = (-\infty, +\infty)\).
- Equation form: \[ y = 3\ln(4x - 6) - 2 \Rightarrow \ln(4x - 6) = \frac{y + 2}{3} \]
- Exponential form: \[ 4x - 6 = e^{(y + 2)/3} \]
- Solve for \(x\): \[ x = \frac{1}{4} e^{(y + 2)/3} + \frac{3}{2} \]
- Inverse function: \[ f^{-1}(x) = \frac{1}{4} e^{(x + 2)/3} + \frac{3}{2} \]
- Domain and range of \(f^{-1}\): \(\text{Domain}(f^{-1}) = (-\infty, +\infty)\), \(\text{Range}(f^{-1}) = \left(\frac{3}{2}, +\infty\right)\)
Example 3
Find the inverse, its domain, and range of:
\[ f(x) = -\ln(x^2 - 4) - 5, \quad x < -2 \]
Solution
- Domain of \(f\): \((-\infty, -2)\), Range: \((-\infty, +\infty)\).
- Equation form: \[ y = -\ln(x^2 - 4) - 5 \Rightarrow \ln(x^2 - 4) = -y - 5 \]
- Exponential form: \[ x^2 - 4 = e^{-y - 5} \Rightarrow x = \pm \sqrt{e^{-y - 5} + 4} \]
- Since \(x < -2\), choose the negative root: \[ x = - \sqrt{e^{-y - 5} + 4} \]
- Inverse function: \[ f^{-1}(x) = - \sqrt{e^{-x - 5} + 4} \]
- Domain and range of \(f^{-1}\): \(\text{Domain}(f^{-1}) = (-\infty, +\infty)\), \(\text{Range}(f^{-1}) = (-\infty, -2)\)
Exercises
Find the inverse, domain, and range of:
- \[ f(x) = - \ln(-x + 4) - 6 \]
- \[ g(x) = \ln(x^2 - 1) - 3, \quad x > 1 \]
Answers
- \[ f^{-1}(x) = - e^{-x - 6} + 4, \quad \text{Domain: } (-\infty, +\infty), \quad \text{Range: } (-\infty, 4) \]
- \[ g^{-1}(x) = \sqrt{1 + e^{x + 3}}, \quad \text{Domain: } (-\infty, +\infty), \quad \text{Range: } (1, +\infty) \]
More References