Math Problem Statement
Solution
The question asks us to identify which graph matches the equation . Let’s analyze it step by step:
Key Properties of :
- Domain: (logs are undefined for ).
- Range: (the log function can take any real value).
- Intercept: When , . Thus, the graph passes through the point .
- Behavior:
- For , (log values are negative for inputs between 0 and 1).
- For , and the graph grows slowly.
- Asymptote: A vertical asymptote exists at .
Matching with Graphs:
- The correct graph should:
- Start from the vertical asymptote at .
- Pass through .
- Be positive for and negative for .
From the image:
- The second graph (choice 2) meets all these properties:
- Vertical asymptote at ,
- Passing through ,
- Negative for and positive for .
Final Answer:
The graph in choice 2 represents .
Would you like a detailed explanation of the logarithmic function or further clarification? Here are five questions to deepen your understanding:
- What happens to the graph of if the base of the logarithm changes (e.g., )?
- How do we solve equations involving algebraically?
- What is the relationship between and its inverse function ?
- How do shifts or transformations (e.g., ) affect the graph?
- What practical applications involve logarithmic functions like ?
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
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