# How do you find the domain and range of y=log(2x-3)/(x-5) ?

Jun 24, 2018

The domain is $x \in \left(\frac{3}{2} , 5\right) \cup \left(5 , + \infty\right)$. The range is $y \in \mathbb{R}$

#### Explanation:

The function is

$y = \ln \frac{2 x - 3}{x - 5}$

For the domain, there are $2$ points to consider

$\left\{\begin{matrix}2 x - 3 > 0 \\ x - 5 \ne 0\end{matrix}\right.$

$\implies$, $\left\{\begin{matrix}x > \frac{3}{2} \\ x \ne 5\end{matrix}\right.$

Therefore,

The domain is $x \in \left(\frac{3}{2} , 5\right) \cup \left(5 , + \infty\right)$

For the range, calculate the following limits

${\lim}_{x \to \frac{3}{2}} y = + \infty$

${\lim}_{x \to {5}^{-}} y = - \infty$

${\lim}_{x \to {5}^{+}} y = + \infty$

${\lim}_{x \to \infty} y = {0}^{+}$

The range is $y \in \mathbb{R}$

graph{ln(2x-3)/(x-5) [-5.73, 16.77, -4.24, 7.01]}