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AP Precalculus -2.12 Logarithmic Function Manipulation- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -2.12 Logarithmic Function Manipulation- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -2.12 Logarithmic Function Manipulation- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

The functions \( f \) and \( g \) are given by \( f(x) = 4^{(5x-1)} \) and \( g(x) = 8^{(x/4)} \). When solving the equation \( f(x) = g(x) \), the functions can be rewritten in equivalent forms so that the equation can be solved without the use of technology. Which of the following are equivalent definitions of \( f \) and \( g \) that aid in solving \( f(x) = g(x) \) without the use of technology?
(A) \( f(x) = 2^{(\log_2 4 \cdot (5x-1))} \) and \( g(x) = 2^{(\log_2 8 \cdot (x/4))} \)
(B) \( f(x) = 2^{(\log_2 8 \cdot (5x-1))} \) and \( g(x) = 2^{(\log_2 4 \cdot (x/4))} \)
(C) \( f(x) = 4^{(\log_2 4 – (5x-1))} \) and \( g(x) = 8^{(\log_2 8 \cdot (x/4))} \)
(D) \( f(x) = 2 \cdot 4^{(\log_2 4 – (5x-1))} \) and \( g(x) = 8^{(\log_2 8 \cdot (x/4))} \)
▶️ Answer/Explanation
Detailed solution

Rewrite \( 4 \) and \( 8 \) as powers of \( 2 \):
\( 4 = 2^2 \), \( 8 = 2^3 \).
So \( f(x) = (2^2)^{5x-1} = 2^{2(5x-1)} = 2^{10x-2} \).
Also \( g(x) = (2^3)^{x/4} = 2^{3x/4} \).
Comparing with the options, \( \log_2 4 = 2 \) and \( \log_2 8 = 3 \).
Thus \( f(x) = 2^{(\log_2 4 \cdot (5x-1))} \) and \( g(x) = 2^{(\log_2 8 \cdot (x/4))} \).
This matches option (A).
✅ Answer: (A)

Question 

To solve the equation \( \log_8(x – 3) + \log_8(x + 4) = 1 \), one method is to apply the properties of logarithms to write a new equation that can be used to identify possible solutions. Of the following, which is such an equation?
(A) \( 2x + 1 = 8 \)
(B) \( \frac{x-3}{x+4} = 8 \)
(C) \( x^2 – 12 = 8 \)
(D) \( x^2 + x – 12 = 8 \)
▶️ Answer/Explanation
Detailed solution

Combine logs: \( \log_8[(x-3)(x+4)] = 1 \).
Convert to exponential: \( (x-3)(x+4) = 8^1 \).
Expand: \( x^2 + x – 12 = 8 \).
✅ Answer: (D)

Question 

An equation involves the expression \( \log_9(27^x) \), which is equivalent to a rational multiple of \( x \). By rewriting the expression in an equivalent form, the value of the rational number can be determined without use of a calculator or complicated calculations. Which of the following is an equivalent expression that satisfies this requirement?
(A) \( x \ln\left(\frac{27}{9}\right) \)
(B) \( x \log_3\left(\frac{27}{9}\right) \)
(C) \( \frac{x \ln 27}{\ln 9} \)
(D) \( \frac{x \log_3 27}{\log_3 9} \)
▶️ Answer/Explanation
Detailed solution

Change to base 3: \( \log_9(27^x) = \frac{\log_3(27^x)}{\log_3 9} = \frac{x \log_3 27}{\log_3 9} \).
Since \( \log_3 27 = 3 \) and \( \log_3 9 = 2 \), the expression simplifies to \( \frac{3}{2}x \).
✅ Answer: (D)

Question 

If \( m = \log_3 81 \), which of the following is also true?
(A) \( 3m = 81 \)
(B) \( 3^m = 81 \)
(C) \( \sqrt[3]{m} = 81 \)
(D) \( \sqrt[3]{81} = m \)
▶️ Answer/Explanation
Detailed solution

By definition of logarithm: \( m = \log_3 81 \) means \( 3^m = 81 \).
✅ Answer: (B)

Question 

The function \( f \) is given by \( f(x) = 9 \cdot 25^x \). Which of the following is an equivalent form for \( f(x) \)?
(A) \( f(x) = 3 \cdot 5^{(x/2)} \)
(B) \( f(x) = 3 \cdot 5^{2x} \)
(C) \( f(x) = 9 \cdot 5^{(x/2)} \)
(D) \( f(x) = 9 \cdot 5^{2x} \)
▶️ Answer/Explanation
Detailed solution

\( 25 = 5^2 \), so \( 25^x = (5^2)^x = 5^{2x} \).
Thus \( f(x) = 9 \cdot 5^{2x} \).
✅ Answer: (D)

Question 

The function \( f \) is given by \( f(x) = \log_2 x \). Which of the following is equivalent to \( f(7) \)?
(A) \( \log_{10} \left( \frac{7}{2} \right) \)
(B) \( \frac{\log_{10} 2}{\log_{10} 7} \)
(C) \( \frac{\log_7 10}{\log_2 10} \)
(D) \( \frac{\log_3 7}{\log_3 2} \)
▶️ Answer/Explanation
Detailed solution

We have \( f(7) = \log_2 7 \). Using the change of base formula for logarithms:
\( \log_b x = \frac{\log_a x}{\log_a b} \), where \( a > 0 \) and \( a \neq 1 \).
Applying this to \( \log_2 7 \), we can choose any valid base \( a \).
For option (D): \( \frac{\log_3 7}{\log_3 2} = \log_2 7 \), which matches \( f(7) \).
✅ Answer: (D)

Question 

The function \( g \) is given by \( g(x) = \log_7 x \), and the function \( h \) is given by \( h(x) = \log_{49} x \). Which of the following describes the relationships between \( g \) and \( h \)?
(A) For equal input values, the output values of \( h \) are half the output values of \( g \).
(B) For equal input values, the output values of \( h \) are twice the output values of \( g \).
(C) For equal input values, the output values of \( h \) are the square of the output values of \( g \).
(D) For equal input values, the output values of \( h \) are the square root of the output values of \( g \).
▶️ Answer/Explanation
Detailed solution

Using the change of base formula:
\( h(x) = \log_{49} x = \frac{\log_7 x}{\log_7 49} \).
Since \( 49 = 7^2 \), \( \log_7 49 = 2 \).
Thus, \( h(x) = \frac{\log_7 x}{2} = \frac{g(x)}{2} \).
Therefore, for the same input \( x \), the output of \( h \) is half the output of \( g \).
✅ Answer: (A)

Question 

The function \( h \) is given by \( h(x) = \log_3 x \). Which of the following is equivalent to the expression \( 2 \cdot h(w) + h(p) \), where \( w \) and \( p \) are values in the domain of \( h \)?
(A) \( \log_3 ((wp)^2) \)
(B) \( (\log_3 w)^2 \cdot (\log_3 p) \)
(C) \( \log_3 (w^2p) \)
(D) \( \log_3 (2wp) \)
▶️ Answer/Explanation
Detailed solution

Given \( h(x) = \log_3 x \), so \( h(w) = \log_3 w \) and \( h(p) = \log_3 p \).
The expression becomes:
\( 2 \cdot h(w) + h(p) = 2 \log_3 w + \log_3 p \).
Using the power property \( a \log_b m = \log_b (m^a) \) and the product property \( \log_b m + \log_b n = \log_b (mn) \):
\( 2 \log_3 w = \log_3 (w^2) \)
\( \log_3 (w^2) + \log_3 p = \log_3 (w^2p) \).
✅ Answer: (C)

Question 

The function \( f \) is given by \( f(x) = \log_{10} x \). The function \( g \) is given by \( g(x) = \log_{10} (x^3) \). Which of the following describes a transformation for which the graph of \( g \) is the image of the graph of \( f \)?
(A) A vertical dilation by a factor of 3
(B) A vertical dilation by a factor of \(\frac{1}{3}\)
(C) A horizontal dilation by a factor of 3
(D) A horizontal dilation by a factor of \(\frac{1}{3}\)
▶️ Answer/Explanation
Detailed solution

Using the power property for logarithms:
\( g(x) = \log_{10} (x^3) = 3 \log_{10} x \).
Since \( f(x) = \log_{10} x \), we have \( g(x) = 3f(x) \).
Multiplying the output of \( f \) by 3 corresponds to a vertical dilation (stretch) by a factor of 3.
✅ Answer: (A)

Question 

The function \( g \) is given by \( g(x) = a \log_b c \), where \( a \), \( b \), and \( c \) are positive integers. Which of the following is an equivalent representation of \( g(x) \)?
(A) \( \log_b(c^a) \)
(B) \( (\log_b c)^a \)
(C) \( \log_b(c^{(1/a)}) \)
(D) \( a \log_{10} b + a \log_{10} c \)
▶️ Answer/Explanation
Detailed solution

Using the power property for logarithms: \( a \log_b c = \log_b (c^a) \).
Therefore, \( g(x) = \log_b (c^a) \).
This matches option (A).
✅ Answer: (A)

Question 

Let \( x \) and \( y \) be positive constants. Which of the following is equivalent to \( 2 \ln x – 3 \ln y \)?
(A) \( \ln\left(\frac{x^2}{y^3}\right) \)
(B) \( \ln(x^2 y^3) \)
(C) \( \ln(2x – 3y) \)
(D) \( \ln\left(\frac{2x}{3y}\right) \)
▶️ Answer/Explanation
Detailed solution

Using logarithm properties:
\( 2 \ln x = \ln(x^2) \)
\( 3 \ln y = \ln(y^3) \)
Subtraction of logs corresponds to division:
\( 2 \ln x – 3 \ln y = \ln(x^2) – \ln(y^3) = \ln\left(\frac{x^2}{y^3}\right) \).
✅ Answer: (A)

Question 

Let \( k, w, \) and \( z \) be positive constants. Which of the following is equivalent to \( \log_{10}\left(\frac{kz}{w^2}\right) \)?
(A) \( \log_{10}(k + z) – \log_{10}(2w) \)
(B) \( \log_{10}k + \log_{10}z – 2\log_{10}w \)
(C) \( \log_{10}k + \log_{10}z – \frac{1}{2}\log_{10}w \)
(D) \( \log_{10}k – \log_{10}z + 2\log_{10}w \)
▶️ Answer/Explanation
Detailed solution

Apply logarithm properties step by step:
\( \log_{10}\left(\frac{kz}{w^2}\right) = \log_{10}(kz) – \log_{10}(w^2) \) (quotient rule)
\( = \left(\log_{10}k + \log_{10}z\right) – 2\log_{10}w \) (product rule and power rule).
This matches option (B).
✅ Answer: (B)

Question 

The function \( f \) is given by \( f(x) = a \cdot c^x \), where \( a > 0 \) and \( c > 1 \). Which of the following is true about the values of constants \( m \) and \( b \) in the equation \( \ln(f(x)) = mx + b \)?
(A) \( m > 0 \) because \( \ln c > 0 \); \( b \) can be any real number because \( \ln a \) can be any real number.
(B) \( m > 0 \) because \( \ln c > 0 \); \( b > 0 \) because \( \ln a > 0 \).
(C) \( m \) can be any real number because \( \ln c \) can be any real number; \( b \) can be any real number because \( \ln a \) can be any real number.
(D) \( m \) can be any real number because \( \ln c \) can be any real number; \( b > 0 \) because \( \ln a > 0 \).
▶️ Answer/Explanation
Detailed solution

Given \( f(x) = a \cdot c^x \), take the natural log of both sides:
\( \ln(f(x)) = \ln(a \cdot c^x) = \ln a + x \ln c \).
This is in the form \( \ln(f(x)) = mx + b \) with \( m = \ln c \) and \( b = \ln a \).
Since \( c > 1 \), \( \ln c > 0 \), so \( m > 0 \).
Since \( a > 0 \), \( \ln a \) can be any real number (positive, negative, or zero).
Thus, \( b \) can be any real number.
✅ Answer: (A)

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