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AP Precalculus -2.14 Logarithmic Function Modeling- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -2.14 Logarithmic Function Modeling- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -2.14 Logarithmic Function Modeling- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

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?
\(x\)5678
\(\ln y\)36912
(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 \(x\).
▶️ Answer/Explanation
Detailed solution

Both \(x\) and \(\ln y\) increase by a constant difference: \(x\) increases by 1, and \(\ln y\) increases by 3. This means \(\ln y\) is a linear function of \(x\). Let \(\ln y = mx + b\). Using points (5,3) and (6,6), slope \(m = 3\), and equation \(\ln y = 3x – 12\). Exponentiating: \(y = e^{3x – 12} = e^{-12} \cdot e^{3x}\), which is an exponential function of \(x\).
✅ Answer: (C)

Question 

Consider the function \(f\). The table gives values of \(\log_3(f(x))\) for selected values of \(x\). Which of the following is a graph of \(y = f(x)\)?
\(x\)0123
\(\log_3(f(x))\)2345

▶️ Answer/Explanation
Detailed solution

From the table, \(\log_3(f(x))\) increases by 1 as \(x\) increases by 1, so \(\log_3(f(x))\) is linear: \(\log_3(f(x)) = x + 2\).
Solve for \(f(x)\): \(f(x) = 3^{x+2} = 3^x \cdot 3^2 = 9 \cdot 3^x\).
This is an exponential function with base 3 and initial value 9 when \(x = 0\). Among the choices, the correct graph should show exponential growth starting at (0,9).
✅ Answer: (D)

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