Question

the table gives ordered pairs $(x, ln y)$. for the function $y = f(x)$, which of the following statements about $f$ is supported by the data in the table?
a the function $f$ is logarithmic because the values of $x$ and the values of $ln y$ both form arithmetic sequences.
b the function $f$ is linear because the values in each column form an arithmetic sequence.
c the function $f$ is exponential because the values of $x$ and the values of $ln y$ both form arithmetic sequences.
d the function $f$ is exponential because the values of $ln y$ increase faster than the values of $x$.

Explanation:

Step1: Check sequences for \( x \)

\( x \): 5,6,7,8 (common difference 1, arithmetic sequence)

Step2: Check sequences for \( \ln y \)

\( \ln y \):3,6,9,12 (common difference 3, arithmetic sequence)

Step3: Relate to function type

If \( \ln y = mx + b \) (linear in \( x \)), then \( y = e^{mx + b} = e^b e^{mx} \), which is exponential. So \( f(x) \) is exponential.

Step4: Evaluate options

Only option C correctly states this relationship.

Answer:

C. The function \( f \) is exponential because the values of \( x \) and the values of \( \ln y \) both form arithmetic sequences.