Math Problem Statement
he table represents select input-output values of a logarithmic function.
x 1 2.5 6.25 15.625 39.0625 f (x) 0 2 4 6 8
Construct the logarithmic function to represent the input-output values.
Solution
To find the logarithmic function that represents the given input-output values, we need to identify the pattern and write the function in the general logarithmic form.
General Form of a Logarithmic Function:
The general form of a logarithmic function is: Where:
- is a constant that affects the amplitude or scale,
- is the base of the logarithm, and
- is the input value.
Step 1: Recognize the Pattern in the Input Values
Looking at the values of in the table:
Notice that each subsequent value is multiplied by to get the next value in the sequence:
So, the input values follow a geometric progression with a common ratio of 2.5.
Step 2: Check the Output Values
The corresponding output values are:
This suggests that the function's output increases by 2 as the input value increases by a factor of 2.5.
Step 3: Find the Base and Scaling Constant
We now hypothesize that the logarithmic function may have the form: Given that the output increases by 2 as the input increases by a factor of 2.5, it suggests that .
Let’s verify this by finding the constant .
At , we know that . So, we substitute into the equation: Since , this holds true, confirming the form is correct.
Step 4: Solve for Using Another Point
Let’s use and : Since , this simplifies to: Thus, .
Step 5: Write the Final Function
Now that we have determined and , the logarithmic function that represents the given input-output values is:
Final Function:
Would you like more details or a breakdown of any specific step? Here are 5 questions that could expand on this:
- How does the choice of base in the logarithmic function affect its graph?
- What are the properties of logarithmic functions and how can they be used in real-world applications?
- How can we graph the function ?
- What happens if we change the base to 10 or to in this logarithmic function?
- How can we transform the logarithmic function to make the graph steeper or flatter?
Tip: The base of a logarithm determines how quickly the function grows. A larger base results in slower growth, while a smaller base results in faster growth.
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Math Problem Analysis
Mathematical Concepts
Logarithmic Functions
Exponential Growth
Geometric Progression
Formulas
f(x) = a log_b(x)
Theorems
Logarithmic Function Properties
Logarithmic Function Scaling
Suitable Grade Level
Grades 10-12
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