Math Problem Statement

Which of the following could represent the graph in the xy-plane of y = log2 x?

Solution

The question asks us to identify which graph matches the equation y=log2xy = \log_2{x}. Let’s analyze it step by step:

Key Properties of y=log2xy = \log_2{x}:

  1. Domain: x>0x > 0 (logs are undefined for x0x \leq 0).
  2. Range: (,)(-\infty, \infty) (the log function can take any real value).
  3. Intercept: When x=1x = 1, y=log21=0y = \log_2{1} = 0. Thus, the graph passes through the point (1,0)(1, 0).
  4. Behavior:
    • For 0<x<10 < x < 1, y<0y < 0 (log values are negative for inputs between 0 and 1).
    • For x>1x > 1, y>0y > 0 and the graph grows slowly.
  5. Asymptote: A vertical asymptote exists at x=0x = 0.

Matching with Graphs:

  • The correct graph should:
    • Start from the vertical asymptote at x=0x = 0.
    • Pass through (1,0)(1, 0).
    • Be positive for x>1x > 1 and negative for 0<x<10 < x < 1.

From the image:

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

Final Answer:

The graph in choice 2 represents y=log2xy = \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=log2xy = \log_2{x} if the base of the logarithm changes (e.g., log10x\log_{10}{x})?
  2. How do we solve equations involving log2x\log_2{x} algebraically?
  3. What is the relationship between y=log2xy = \log_2{x} and its inverse function y=2xy = 2^x?
  4. How do shifts or transformations (e.g., y=log2(x2)y = \log_2(x - 2)) affect the graph?
  5. What practical applications involve logarithmic functions like y=log2xy = \log_2{x}?

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

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Graph Analysis

Formulas

y = log_b x

Theorems

Properties of Logarithmic Functions

Suitable Grade Level

Grades 10-12