The question asks us to identify which graph matches the equation $y = \log_2{x}$. Let’s analyze it step by step:

Key Properties of $y = \log_2{x}$:

1. Domain: $x > 0$ (logs are undefined for $x \leq 0$).
2. Range: $(-\infty, \infty)$ (the log function can take any real value).
3. Intercept: When $x = 1$, $y = \log_2{1} = 0$. Thus, the graph passes through the point $(1, 0)$.
4. Behavior:
   • For $0 < x < 1$, $y < 0$ (log values are negative for inputs between 0 and 1).
   • For $x > 1$, $y > 0$ and the graph grows slowly.
5. Asymptote: A vertical asymptote exists at $x = 0$.

Matching with Graphs:

• The correct graph should:
  • Start from the vertical asymptote at $x = 0$.
  • Pass through $(1, 0)$.
  • Be positive for $x > 1$ and negative for $0 < x < 1$.

From the image:

• The second graph (choice 2) meets all these properties:
  • Vertical asymptote at $x = 0$,
  • Passing through $(1, 0)$,
  • Negative for $x \in (0, 1)$ and positive for $x > 1$.

Final Answer:

The graph in choice 2 represents $y = \log_2{x}$.

Would you like a detailed explanation of the logarithmic function or further clarification? Here are five questions to deepen your understanding:

1. What happens to the graph of $y = \log_2{x}$ if the base of the logarithm changes (e.g., $\log_{10}{x}$)?
2. How do we solve equations involving $\log_2{x}$ algebraically?
3. What is the relationship between $y = \log_2{x}$ and its inverse function $y = 2^x$?
4. How do shifts or transformations (e.g., $y = \log_2(x - 2)$) affect the graph?
5. What practical applications involve logarithmic functions like $y = \log_2{x}$?

Tip: Always check the domain and key points (like intercepts) when analyzing a function’s graph.