Question

match the graph with its equation

1. $y = \log_{5}(x)-1$
2. $y = \log_{5}(x - 1)+1$
3. $y = \log_{5}(x^{2})+1$
4. $y = \log_{5}(x + 1)+1$

Explanation:

Step1: Identify domain of parent log

Parent function: $y=\log_5(x)$, domain $x>0$.

Step2: Analyze graph's domain

The shown graph is defined only for $x>1$ (shifted right 1 unit).

Step3: Check equation domains

• $y=\log_5(x)-1$: domain $x>0$ (does not match)
• $y=\log_5(x-1)+1$: domain $x-1>0 \implies x>1$ (matches)
• $y=\log_5(x^2)+1$: domain $x

eq0$ (does not match)

• $y=\log_5(x+1)+1$: domain $x>-1$ (does not match)

Step4: Verify vertical shift

At $x=2$, $y=\log_5(2-1)+1=\log_5(1)+1=0+1=1$, which aligns with the graph's behavior.

Answer:

1. $y = \log_5 (x - 1) + 1$