# How do you find the domain and range of h(x)=ln(x-6)?

Feb 3, 2015

The answers are: $D \left(6 , + \infty\right)$ and $R \left(- \infty , + \infty\right)$.

The domain of the function $y = \ln f \left(x\right)$ is: $f \left(x\right) > 0$.

So:

$x - 6 > 0 \Rightarrow x > 6$ or we can write: $D = \left(6 , + \infty\right)$

The range of a function is the domain of the inverse function. The inverse function of the logarithmic function is the exponential function.

So (using the method to find the inverse function, that is: exchange $x$ with $y$ and finding $y$):

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

that has domain $\left(- \infty , + \infty\right)$.

The function is:

graph{ln(x-6) [-2, 15, -5, 5]}