Question

choose the graph of $y = \log_{3}(x - 1)$

Explanation:

Step1: Find vertical asymptote

For $y=\log_3(x-1)$, set argument to 0:
$x-1=0 \implies x=1$

Step2: Find domain

Logarithm argument > 0:
$x-1>0 \implies x>1$

Step3: Find key point

Set $x-1=3^0=1$, so $x=2$, $y=0$. The graph passes through $(2,0)$.

Step4: Analyze function behavior

Since base $3>1$, the function is increasing for $x>1$. As $x\to1^+$, $y\to-\infty$; as $x\to+\infty$, $y\to+\infty$.

Answer:

The correct graph is the top-left one (the increasing curve with vertical asymptote $x=1$, passing through $(2,0)$, defined only for $x>1$).