Home / AP® Exam / AP® PreCalculus / AP® Precalculus

AP Precalculus -2.11 Logarithmic Functions- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -2.11 Logarithmic Functions- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -2.11 Logarithmic Functions- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

The function \( f \) is given by \( f(x) = \log_3 x \). Which of the following could be the graph of \( y = f^{-1}(x) \)?

▶️ Answer/Explanation
Detailed solution

\( f^{-1}(x) = 3^x \), which is an exponential function with base >1.
It passes through (0,1), increases, and has a horizontal asymptote at \( y=0 \) as \( x \to -\infty \).
Only one of the graphs matches this exponential shape.
✅ Answer: (C)

Question 

The function \( f \) is an increasing function such that every time the output values of the function \( f \) increase by 1, the corresponding input values multiply by 4. Which of the following could define \( f(x) \)?
(A) \( x^4 \)
(B) \( 4^x \)
(C) \( \sqrt[4]{x} \)
(D) \( \log_4 x \)
▶️ Answer/Explanation
Detailed solution

If output increases by 1 when input multiplies by 4, then \( f(4x) = f(x) + 1 \).
This is a property of logarithmic functions: \( \log_b(kx) = \log_b k + \log_b x \).
With \( b=4 \), \( \log_4(4x) = \log_4 4 + \log_4 x = 1 + \log_4 x \).
✅ Answer: (D)

Question 

The function \( f \) is given by \( f(x) = 2 \log_5 x \). Which of the following describes \( f \)?
(A) \( f \) is an increasing function that increases at an increasing rate.
(B) \( f \) is an increasing function that increases at a decreasing rate.
(C) \( f \) is a decreasing function that decreases at an increasing rate.
(D) \( f \) is a decreasing function that decreases at a decreasing rate.
▶️ Answer/Explanation
Detailed solution

\( f(x) = 2 \log_5 x \) is a logarithmic function with base >1, so it is increasing.
Derivative \( f'(x) = \frac{2}{x \ln 5} \) which is positive but decreasing as \( x \) increases, so rate of increase slows down.
Graph is concave down ⇒ increases at a decreasing rate.
✅ Answer: (B)

Question 

Which of the following could describe a single logarithmic function \( f \)?
(A) \( \lim_{x \to 0^+} f(x) = -\infty \) and \( \lim_{x \to \infty} f(x) = -\infty \)
(B) \( \lim_{x \to 0^+} f(x) = -\infty \) and \( \lim_{x \to \infty} f(x) = k \), where \( k \) is a positive constant
(C) \( \lim_{x \to 0^+} f(x) = \infty \) and \( \lim_{x \to \infty} f(x) = 0 \)
(D) \( \lim_{x \to 0^+} f(x) = \infty \) and \( \lim_{x \to \infty} f(x) = -\infty \)
▶️ Answer/Explanation
Detailed solution

Typical log function \( \log_b x \) with \( b>1 \) satisfies \( \lim_{x \to 0^+} = -\infty \), \( \lim_{x \to \infty} = \infty \).
But if we take \( f(x) = -\log_b x \) (still logarithmic), then \( \lim_{x \to 0^+} = \infty \), \( \lim_{x \to \infty} = -\infty \).
This matches (D).
✅ Answer: (D)

Question 

The logarithmic function \( f \) is defined by \( f(x) = \log_3 x \) on a domain of \( 0 < x \leq 9 \). Which of the following is true of \( f \)?
(A) \( f \) has both a maximum and a minimum value.
(B) \( f \) has a maximum value, but no minimum value.
(C) \( f \) has a minimum value, but no maximum value.
(D) \( f \) has neither a minimum value nor a maximum value.
▶️ Answer/Explanation
Detailed solution

\( f \) is increasing on \( (0, 9] \).
As \( x \to 0^+ \), \( f(x) \to -\infty \), so no minimum.
At \( x = 9 \), \( f(9) = \log_3 9 = 2 \), which is the maximum on the closed interval at the right endpoint.
✅ Answer: (B)

Question 

The function \( f \) is given by \( f(x) = \ln x \). Which of the following describes input values for which the output values of \( f \) are integers?
(A) Integer powers of \( e \)
(B) Integer powers of 10
(C) Integers raised to the power \( e \)
(D) Integers raised to the power 10
▶️ Answer/Explanation
Detailed solution

The natural logarithm \( \ln x \) is the logarithm with base \( e \), so \( \ln(e^k) = k \) for any real number \( k \).
If \( k \) is an integer, then \( \ln(e^k) \) is an integer.
Therefore, input values that are integer powers of \( e \) produce integer outputs.
✅ Answer: (A)

Question 

The function \( f \) is logarithmic, and the points \( (2,1) \) and \( (4,2) \) are on the graph of \( f \) in the \( xy \)-plane. Which of the following could define \( f(x) \)?
(A) \( \log_4 x \)
(B) \( 2\log_2 x \)
(C) \( 2\log_4 x \)
(D) \( \log_4(x + 2) \)
▶️ Answer/Explanation
Detailed solution

Check each option against the two given points \( (2,1) \) and \( (4,2) \).
(C) \( f(x) = 2 \log_4 x \):
For \( x = 2 \): \( f(2) = 2 \log_4 2 = 2 \cdot \frac{1}{2} = 1 \) ✅
For \( x = 4 \): \( f(4) = 2 \log_4 4 = 2 \cdot 1 = 2 \) ✅
Both points satisfy the equation.
✅ Answer: (C)

Scroll to Top