Math Problem Statement
Question 8(Multiple Choice Worth 1 points) (Logarithmic Function Context and Data Modeling MC)
calculator Write the logarithmic function h(x) that passes through the coordinates (5, 6.2) and (25, 12.4). h(x) = 5log6.2 x h(x) = log6.2 x h(x) = 6.2log5 x h(x) = log5 x
Solution
To find the logarithmic function that passes through the points and , we use the general form of a logarithmic function:
Where is a constant and is the base of the logarithm.
Step 1: Use the two points to set up a system of equations.
We are given two points: and .
Substitute these values into the general logarithmic function form.
For the first point :
For the second point :
Step 2: Solve the system of equations.
We now have two equations:
By dividing the second equation by the first:
This simplifies to:
Using the logarithmic property , we get:
This simplifies to:
This equation holds true, meaning our assumption about the base being is correct.
Step 3: Determine the form of the function.
Now that we know the base , the logarithmic function is in the form:
Step 4: Find .
Substitute the first point into the equation:
Since , we have:
Thus, .
Final Answer:
The function is:
So, the correct answer is:
h(x) = 6.2 log5 x
Would you like any further clarification or have any questions about the process?
Here are 5 related questions to explore further:
- How do you solve systems of equations involving logarithmic functions?
- How do you solve for the base of a logarithmic function?
- What is the general form of a logarithmic function?
- How does the logarithmic property apply in solving these types of problems?
- How would the function change if the base were different, say ?
Tip: Always check if the base of the logarithm is specified or can be inferred from the data points provided!
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Math Problem Analysis
Mathematical Concepts
Logarithmic Functions
Systems of Equations
Logarithmic Properties
Formulas
h(x) = a log_b(x)
Theorems
Logarithmic property: log_b(x^n) = n log_b(x)
Change of base property
Suitable Grade Level
Grades 9-11
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