# What is the inverse of y = ln(x) + ln(x-6)  ?

Jun 8, 2018

For the inverse to be a function a domain restriction will be required:

$y ' = 3 \pm \sqrt{{e}^{x} + 9}$

#### Explanation:

$y = \ln \left(x\right) + \ln \left(x - 6\right)$

$x = \ln \left(y\right) + \ln \left(y - 6\right)$

Apply rule: $\ln \left(a\right) + \ln \left(b\right) = \ln \left(a b\right)$

$x = \ln \left(y \left(y - 6\right)\right)$

${e}^{x} = {e}^{\ln \left(y \left(y - 6\right)\right)}$

${e}^{x} = y \left(y - 6\right)$

${e}^{x} = {y}^{2} - 6 y$

complete the square:

${e}^{x} + 9 = {y}^{2} - 6 y + 9$

${e}^{x} + 9 = {\left(y - 3\right)}^{2}$

$y - 3 = \pm \sqrt{{e}^{x} + 9}$

$y = 3 \pm \sqrt{{e}^{x} + 9}$