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AP Precalculus -2.13 Exponential and Logarithmic Equations and Inequalities- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -2.13 Exponential and Logarithmic Equations and Inequalities- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -2.13 Exponential and Logarithmic Equations and Inequalities- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

The range of function \( f \) is the positive real numbers. The function \( g \) is given by \( g(x) = \ln(f(x)) \). Solutions to which of the following equations are useful in solving \( g(x) = 2 \)?
(A) \( f(x) = 2 \)
(B) \( f(x) = e^2 \)
(C) \( f(x) = 10^2 \)
(D) \( f(x) = \frac{2}{\ln x} \)
▶️ Answer/Explanation
Detailed solution

\( g(x) = 2 \) ⇒ \( \ln(f(x)) = 2 \) ⇒ \( f(x) = e^2 \) by definition of natural log.
✅ Answer: (B)

Question 

What are all values of \( x \) for which \( \ln(x^3) – \ln x = 4 \)?
(A) \( x = -2 \) and \( x = 2 \)
(B) \( x = -e^2 \) and \( x = e^2 \)
(C) \( x = e^2 \) only
(D) \( x = e^4 \)
▶️ Answer/Explanation
Detailed solution

Combine logs: \( \ln\left(\frac{x^3}{x}\right) = \ln(x^2) = 4 \).
So \( x^2 = e^4 \) ⇒ \( x = \pm e^2 \).
Domain of original: \( \ln(x^3) \) requires \( x>0 \), and \( \ln x \) requires \( x>0 \), so \( x>0 \).
Thus \( x = e^2 \) only.
✅ Answer: (C)

Question 

Consider the functions \( f \) and \( g \) given by \( f(x) = \log_{10}(x – 1) + \log_{10}(x + 3) \) and \( g(x) = \log_{10}(x + 9) \). In the \( xy \)-plane, what are all \( x \)-coordinates of the points of intersection of the graphs of \( f \) and \( g \)?
(A) \( x = 3 \) only
(B) \( x = 7 \)
(C) \( x = -4 \) and \( x = 3 \)
(D) \( x = -7 \) and \( x = -4 \)
▶️ Answer/Explanation
Detailed solution

Combine \( f \): \( f(x) = \log_{10}[(x-1)(x+3)] \).
Set \( f(x) = g(x) \): \( \log_{10}[(x-1)(x+3)] = \log_{10}(x+9) \) ⇒ \( (x-1)(x+3) = x+9 \).
Expand: \( x^2 + 2x – 3 = x + 9 \) ⇒ \( x^2 + x – 12 = 0 \) ⇒ \( (x+4)(x-3) = 0 \) ⇒ \( x = -4 \) or \( x = 3 \).
Domain of \( f \): \( x-1 > 0 \) ⇒ \( x>1 \) and \( x+3>0 \) ⇒ \( x>-3 \), so \( x>1 \).
Thus \( x = -4 \) is extraneous; only \( x=3 \).
✅ Answer: (A)

Question 

The function \( g \) is given by \( g(x) = \ln(3x + 1) – \ln(x^2 + x – 2) \). What are all values of \( x \) for which \( g(x) < 0 \)?
(A) \( (-\infty, -1) \) and \( (3, \infty) \)
(B) \( (-1, 3) \)
(C) \( (1, 3) \) only
(D) \( (3, \infty) \) only
▶️ Answer/Explanation
Detailed solution

\( g(x) < 0 \) ⇒ \( \ln\frac{3x+1}{x^2 + x – 2} < 0 \) ⇒ \( \frac{3x+1}{x^2 + x – 2} < 1 \), given the fraction is positive (domain).
Domain: \( 3x+1 > 0 \) ⇒ \( x > -\frac{1}{3} \), and \( x^2 + x – 2 > 0 \) ⇒ \( (x+2)(x-1) > 0 \) ⇒ \( x < -2 \) or \( x > 1 \).
Intersection: \( x > 1 \).
Within \( x>1 \), solve \( \frac{3x+1}{x^2 + x – 2} < 1 \).
Since denominator > 0, multiply: \( 3x+1 < x^2 + x – 2 \) ⇒ \( 0 < x^2 – 2x – 3 \) ⇒ \( (x-3)(x+1) > 0 \) ⇒ \( x < -1 \) or \( x > 3 \).
Combine with \( x>1 \): \( x > 3 \).
✅ Answer: (D)

Question 

A decibel (dB) is a unit of measure for loudness of sound. The decibel scale is based on sound intensity \( N \), in watts per square meter. A decibel value is given by the function \( d(N) = 10 \log_{10} \left( \frac{N}{10^{-12}} \right) \). Which of the following gives all intensities \( N \), in watts per square meter, for which the decibel value is greater than 140 decibels?
(A) \( N > 14 \cdot 10^{-12} \)
(B) \( N > 2 \)
(C) \( N > 100 \)
(D) \( N > 10^{26} \)
▶️ Answer/Explanation
Detailed solution

\( d(N) > 140 \) ⇒ \( 10 \log_{10} \left( \frac{N}{10^{-12}} \right) > 140 \)
Divide by 10: \( \log_{10} \left( \frac{N}{10^{-12}} \right) > 14 \)
Convert: \( \frac{N}{10^{-12}} > 10^{14} \)
Multiply: \( N > 10^{14} \cdot 10^{-12} = 10^{2} = 100 \).
✅ Answer: (C)

Question 

Consider the functions \( f \) and \( g \) given by \( f(x) = \log_{10}(x-1) + \log_{10}(x+3) \) and \( g(x) = \log_{10}(x+9) \). In the xy-plane, what are all \( x \)-coordinates of the points of intersection of the graphs of \( f \) and \( g \)?
(A) \( x = 3 \) only
(B) \( x = 7 \)
(C) \( x = -4 \) and \( x = 3 \)
(D) \( x = -7 \) and \( x = -4 \)
▶️ Answer/Explanation
Detailed solution

Combine logs for \( f \):
\[ f(x) = \log_{10}[(x-1)(x+3)] = \log_{10}(x^2 + 2x – 3) \]
Set \( f(x) = g(x) \):
\[ \log_{10}(x^2 + 2x – 3) = \log_{10}(x+9) \]
Equate arguments (since logs are same base and function is 1–1):
\[ x^2 + 2x – 3 = x + 9 \]
\[ x^2 + x – 12 = 0 \]
\[ (x+4)(x-3) = 0 \]
\( x = -4 \) or \( x = 3 \).
Check domain: for \( f \), need \( x-1 > 0 \) and \( x+3 > 0 \) ⇒ \( x > 1 \). So \( x = -4 \) is extraneous.
Only \( x = 3 \) works.
✅ Answer: (A)

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