Question

indicate which of the following four graphs is the correct graph of this function.

m(x) = \log_{5}(x - 1)

answer

Explanation:

Step1: Analyze the function's domain

The function is \( m(x)=\log_{5}(x - 1) \). For a logarithmic function \( \log_{a}(u) \), the argument \( u>0 \). So \( x - 1>0\implies x>1 \). So the domain is \( x\in(1,+\infty) \). We can eliminate graphs where the curve is defined for \( x\leq1 \).

Step2: Analyze the behavior of the logarithmic function

The base of the logarithm \( 5>1 \), so the function \( \log_{5}(u) \) is an increasing function (as the base \( a > 1 \) for logarithmic functions \( y=\log_{a}u \) implies increasing). Also, when \( x = 2 \), \( m(2)=\log_{5}(2 - 1)=\log_{5}(1) = 0 \), so the graph passes through \( (2,0) \).

Now, let's check the graphs:

• The first graph: The curve is decreasing and defined for \( x<4 \) (approx), which is not consistent with domain \( x > 1 \) and increasing nature. Eliminate.
• The second graph: The curve is going downwards (decreasing) and has a vertical asymptote, but the domain and increasing nature don't match. Eliminate.
• The third graph: The vertical asymptote seems to be at \( x=- 2 \) (approx), but our domain is \( x>1 \), so the asymptote should be at \( x = 1 \). Eliminate.
• The fourth graph: The vertical asymptote is at \( x = 1 \) (since the curve starts from \( x>1 \)), it is increasing (as \( x \) increases, \( y \) increases slowly), and when \( x = 2 \), \( y = 0 \) (consistent with our calculation). So this graph matches.

Answer:

The bottom - right graph (the fourth graph in the given set of four graphs)