Answer
Using a graphing utility, the graphs of the functions $y_1$ and $y_2$ are as shown.
No. From the domain of $y_1$ which is $x\gt3$ and from the domain of $y_2$ which is $x\gt0\text{ or }x\gt 3$, some numbers are in the domain of one function but not the other.
Work Step by Step
Using a graphing utility, the graphs of the functions $y_1$ and $y_2$ are as shown.
No. The graphs do not show the functions with the same domain.
Finding the domain of $y_1$:
From $\ln x$:
$$x\gt0$$
From $\ln (x-3)$:
$$x-3\gt0$$ $$x\gt3$$
Combining:
$$x\gt0\text{ and }x\gt3$$ $$x\gt3$$
Thus, the domain for $y_1$ is $x\gt3$.
Finding the domain of $y_2$:
$$\frac{x}{x-3}\gt0$$
Finding the intervals:
$$x=0$$ $$x-3=0$$ $$x=3$$
Then, the intervals are:
$$x\lt0,~0\lt x\lt3,~x\gt3$$
Checking for valid solutions:
For $x=-1$:
$$\frac{-1}{-1-3}\gt0$$ $$\frac{1}{4}\gt0~True$$
Thus, $x\lt0$ is an interval for the domain.
At $x=1$:
$$\frac{1}{1-3}\gt0$$ $$-\frac{1}{2}\gt0~False$$
Thus, $0\lt x\lt 3$ is not an interval for the domain.
At $x=4$:
$$\frac{4}{4-3}\gt0$$ $$4\gt0~True$$
Thus, $x\ge3$ is an interval for the domain.
Thus, combining the valid intervals which is the domain for $y_2$:
$$x\lt0\text{ or }x\gt 3$$
Therefore, from the domain of $y_1$ which is $x\gt3$ and from the domain of $y_2$ which is $x\lt0\text{ or }x\gt 3$, some numbers are in the domain of one function but not the other.
