Math Problem Statement
Solution
We are solving this problem where the graph describes a linear relationship between and . Let's analyze it step by step:
1. Observations from the graph:
- The line equation in the plane is a straight line. It passes through:
- (i.e., when , ),
- (i.e., when , ).
This suggests the line has a slope and intercept that we can use to find the relationship.
2. Finding the slope and equation of the line:
The slope of the line is:
The line equation in terms of and is:
Using the point to find the intercept:
Thus, the equation of the line is:
3. Converting the equation into exponential form:
To simplify the relationship, rewrite the logarithms in terms of exponents.
Step 1: Rewrite and in base 2:
Substituting these into the line equation:
Multiply through by 3 to eliminate the denominator:
Step 2: Combine the terms:
Using the properties of logarithms:
Thus:
4. Final Answer:
The correct option is D. .
Would you like further clarification on any of the steps?
Here are five related questions for practice:
- How do you find the equation of a line given two points?
- What are the properties of logarithms used in this problem?
- How do you convert logarithms of different bases?
- How does the slope affect the relationship between variables?
- Can you derive the relationship between and if the slope was different?
Tip: Always use logarithm properties () to simplify equations!
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Math Problem Analysis
Mathematical Concepts
Logarithms
Linear Equations
Exponential Relationships
Change of Base Formula
Formulas
Slope formula: m = (y2 - y1) / (x2 - x1)
Logarithmic properties: log_a(b^c) = c * log_a(b)
Change of base formula: log_a(b) = log_c(b) / log_c(a)
Theorems
Logarithm and exponentiation relationship: log_a(b) = x ↔ a^x = b
Suitable Grade Level
Grades 10-12
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