Which description matches the graph of $y=\log (x+2)$ (base 10)?

• A. An increasing logarithmic curve with vertical asymptote $x=2$ and domain $x>2$.
• B. An increasing logarithmic curve with vertical asymptote $x=0$ and x-intercept at $(1,0)$.
• C. A decreasing logarithmic curve with vertical asymptote $x=-2$ and domain $x<-2$.
• D. An increasing logarithmic curve with vertical asymptote $x=-2$, domain $x>-2$, and x-intercept at $(-1,0)$.

Answer: (D)

Verified step by step guidance

1. Identify the function given: \(y = \log_{10}(x + 2)\). This is a logarithmic function with base 10 and an argument of \((x + 2)\).
2. Determine the domain of the function by setting the argument greater than zero: \(x + 2 > 0\). Solve this inequality to find the domain.
3. Find the vertical asymptote by setting the argument equal to zero: \(x + 2 = 0\). Solve for \(x\) to locate the vertical asymptote.
4. Check the behavior of the logarithmic function: since the base 10 logarithm is an increasing function, the graph will be increasing on its domain.
5. Find the x-intercept by setting \(y = 0\) and solving for \(x\): \$0 = \log_{10}(x + 2)\( implies \)x + 2 = 1\(. Solve for \)x$ to find the x-intercept.