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: f(x)=alogb(x)f(x) = a \log_b(x) Where:

  • aa is a constant that affects the amplitude or scale,
  • bb is the base of the logarithm, and
  • xx is the input value.

Step 1: Recognize the Pattern in the Input Values

Looking at the values of xx in the table:

x=1,2.5,6.25,15.625,39.0625x = 1, 2.5, 6.25, 15.625, 39.0625

Notice that each subsequent value is multiplied by 2.52.5 to get the next value in the sequence: 1×2.5=2.51 \times 2.5 = 2.5 2.5×2.5=6.252.5 \times 2.5 = 6.25 6.25×2.5=15.6256.25 \times 2.5 = 15.625 15.625×2.5=39.062515.625 \times 2.5 = 39.0625

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:

f(1)=0,f(2.5)=2,f(6.25)=4,f(15.625)=6,f(39.0625)=8f(1) = 0, \, f(2.5) = 2, \, f(6.25) = 4, \, f(15.625) = 6, \, f(39.0625) = 8

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: f(x)=alogb(x)f(x) = a \log_b(x) Given that the output increases by 2 as the input increases by a factor of 2.5, it suggests that b=2.5b = 2.5.

Let’s verify this by finding the constant aa.

At x=1x = 1, we know that f(1)=0f(1) = 0. So, we substitute x=1x = 1 into the equation: 0=alog2.5(1)0 = a \log_{2.5}(1) Since log2.5(1)=0\log_{2.5}(1) = 0, this holds true, confirming the form is correct.

Step 4: Solve for aa Using Another Point

Let’s use x=2.5x = 2.5 and f(2.5)=2f(2.5) = 2: 2=alog2.5(2.5)2 = a \log_{2.5}(2.5) Since log2.5(2.5)=1\log_{2.5}(2.5) = 1, this simplifies to: 2=a12 = a \cdot 1 Thus, a=2a = 2.

Step 5: Write the Final Function

Now that we have determined a=2a = 2 and b=2.5b = 2.5, the logarithmic function that represents the given input-output values is: f(x)=2log2.5(x)f(x) = 2 \log_{2.5}(x)

Final Function:

f(x)=2log2.5(x)f(x) = 2 \log_{2.5}(x)


Would you like more details or a breakdown of any specific step? Here are 5 questions that could expand on this:

  1. How does the choice of base in the logarithmic function affect its graph?
  2. What are the properties of logarithmic functions and how can they be used in real-world applications?
  3. How can we graph the function f(x)=2log2.5(x)f(x) = 2 \log_{2.5}(x)?
  4. What happens if we change the base to 10 or to ee in this logarithmic function?
  5. 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