Question

the table above represents a data set for ordered pairs \\((x, \ln y)\\). if \\(y = f(x)\\), which statement below best describes \\(f\\)?

• the \\(x\\) values and \\(\ln y\\) values both form a geometric sequence, therefore \\(f\\) is exponential.
• the \\(x\\) values and \\(\ln y\\) values both form an arithmetic sequence, therefore \\(f\\) is exponential.
• the \\(x\\) values and \\(\ln y\\) values both form a geometric sequence, therefore \\(f\\) is quadratic.
• the \\(x\\) values and \\(\ln y\\) values both form an arithmetic sequence, therefore \\(f\\) is linear.

Explanation:

🆕 New Concept Discovered: Linearizing Data and Semi-Log Models
Using logarithms to identify exponential relationships.

Step 1: Analyze the sequences of $x$ and $\ln y$

Examine the values in the table:

• The $x$-values are: \(1, 2, 3, 4\). Each term increases by adding \(1\). This is an arithmetic sequence.
• The \(\ln y\) values are: \(2, 4, 6, 8\). Each term increases by adding \(2\). This is also an arithmetic sequence.

Step 2: Determine the relationship between $x$ and $\ln y$

Since both sequences are arithmetic and increase by constant differences, there is a linear relationship between \(x\) and \(\ln y\):

$$ \ln y = mx + b $$

Using the points \((1, 2)\) and \((2, 4)\):

• Slope \(m = \frac{4 - 2}{2 - 1} = 2\)
• Equation: \(\ln y = 2x\)

Step 3: Solve for $y$ to find the function type

To find \(y = f(x)\), exponentiate both sides using base \(e\):

$$ e^{\ln y} = e^{2x} $$

$$ y = e^{2x} $$

$$ y = (e^2)^x $$

This is in the form \(y = a \cdot b^x\), which represents an exponential function. Therefore, \(f\) is exponential.

Answer:

The $x$ values and $\ln y$ values both form an arithmetic sequence, therefore $f$ is exponential.