Which of the following statements correctly describes the graph of $y=\log (-x)$ (where $\log$ denotes base-10 logarithm)?

• A. It is the graph of $y=\log \left(x\right)$ shifted left 1 unit, with domain $x>1$ and vertical asymptote $x=1$.
• B. It is the reflection of $y=\log \left(x\right)$ across the $x$-axis, with domain $x>0$ and vertical asymptote $y=0$.
• C. It is the reflection of $y=\log \left(x\right)$ across the origin, with domain $x>0$ and vertical asymptote $x=0$.
• D. It is the reflection of $y=\log \left(x\right)$ across the $y$-axis, with domain $x<0$ and vertical asymptote $x=0$.

Answer: (D)

Verified step by step guidance

1. Recall the basic graph of the logarithmic function \(y = \log(x)\), which has a domain of \(x > 0\) and a vertical asymptote at \(x = 0\).
2. Consider the function \(y = \log(-x)\). To understand its graph, analyze the inside of the logarithm: \(-x\) must be positive, so the domain is \(-x > 0\), which simplifies to \(x < 0\).
3. Since the input to the logarithm is \(-x\), this means the graph of \(y = \log(-x)\) is obtained by reflecting the graph of \(y = \log(x)\) across the \(y\)-axis (because replacing \(x\) by \(-x\) reflects points horizontally).
4. The vertical asymptote of \(y = \log(x)\) is at \(x = 0\). Reflecting across the \(y\)-axis does not change the location of the vertical asymptote, so \(y = \log(-x)\) also has a vertical asymptote at \(x = 0\).
5. Summarize: The graph of \(y = \log(-x)\) is the reflection of \(y = \log(x)\) across the \(y\)-axis, with domain \(x < 0\) and vertical asymptote at \(x = 0\).