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 h(x)h(x) that passes through the points (5,6.2)(5, 6.2) and (25,12.4)(25, 12.4), we use the general form of a logarithmic function:

h(x)=alogb(x)h(x) = a \log_b(x)

Where aa is a constant and bb is the base of the logarithm.

Step 1: Use the two points to set up a system of equations.

We are given two points: (5,6.2)(5, 6.2) and (25,12.4)(25, 12.4).

Substitute these values into the general logarithmic function form.

For the first point (5,6.2)(5, 6.2): 6.2=alogb(5)6.2 = a \log_b(5)

For the second point (25,12.4)(25, 12.4): 12.4=alogb(25)12.4 = a \log_b(25)

Step 2: Solve the system of equations.

We now have two equations: 6.2=alogb(5)6.2 = a \log_b(5) 12.4=alogb(25)12.4 = a \log_b(25)

By dividing the second equation by the first: 12.46.2=alogb(25)alogb(5)\frac{12.4}{6.2} = \frac{a \log_b(25)}{a \log_b(5)}

This simplifies to: 2=logb(25)logb(5)2 = \frac{\log_b(25)}{\log_b(5)}

Using the logarithmic property logb(25)=logb(52)=2logb(5)\log_b(25) = \log_b(5^2) = 2 \log_b(5), we get: 2=2logb(5)logb(5)2 = \frac{2 \log_b(5)}{\log_b(5)}

This simplifies to: 2=22 = 2

This equation holds true, meaning our assumption about the base being b=5b = 5 is correct.

Step 3: Determine the form of the function.

Now that we know the base b=5b = 5, the logarithmic function is in the form: h(x)=alog5(x)h(x) = a \log_5(x)

Step 4: Find aa.

Substitute the first point (5,6.2)(5, 6.2) into the equation: 6.2=alog5(5)6.2 = a \log_5(5)

Since log5(5)=1\log_5(5) = 1, we have: 6.2=a6.2 = a

Thus, a=6.2a = 6.2.

Final Answer:

The function is: h(x)=6.2log5(x)h(x) = 6.2 \log_5(x)

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:

  1. How do you solve systems of equations involving logarithmic functions?
  2. How do you solve for the base of a logarithmic function?
  3. What is the general form of a logarithmic function?
  4. How does the logarithmic property logb(xn)=nlogb(x)\log_b(x^n) = n \log_b(x) apply in solving these types of problems?
  5. How would the function change if the base were different, say b=10b = 10?

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