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AP Precalculus -2.15 Semi-log Plots- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -2.15 Semi-log Plots- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -2.15 Semi-log Plots- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

In a semi-log plot, which of the following pairs of functions appear linear as parallel lines?
(A) \( f(x) = 2x \) and \( g(x) = 2x + 3 \)
(B) \( f(x) = x^2 \) and \( g(x) = 3x^2 \)
(C) \( f(x) = 2^x \) and \( g(x) = 3 \cdot 2^x \)
(D) \( f(x) = \ln(2x) \) and \( g(x) = 3 \ln(2x) \)
▶️ Answer/Explanation
Detailed solution

A semi-log plot has a logarithmic scale on the vertical axis and a linear scale on the horizontal axis. Exponential functions \( y = k \cdot a^x \) appear as straight lines because \( \log y = \log k + x \log a \), which is linear in \( x \).
For \( f(x) = 2^x \) and \( g(x) = 3 \cdot 2^x \):
In a semi-log plot, \( \log f(x) = x \log 2 \) and \( \log g(x) = \log 3 + x \log 2 \).
Both are lines with slope \( \log 2 \), but different intercepts, so they appear as parallel lines.
✅ Answer: (C)

Question 

 
 
 
 
 
 
 
 
 
 
 
The number of thousands of people that have visited a new website is recorded every 10 days for 60 days. These data are used to produce a semi-log plot as shown. The function \( N \) gives the number of thousands of people that have visited the website for day \( t \). Which of the following could define \( N(t) \)?
(A) \( \frac{1}{2}t \)
(B) \( \frac{1}{10}t + 5 \)
(C) \( 2.5 \cdot 2^{(t/10)} \)
(D) \( 3 + 2^{(t/10)} \)
▶️ Answer/Explanation
Detailed solution

A semi-log plot appears linear when the data follow an exponential model. In such a plot, a straight line corresponds to exponential growth of the form \( N(t) = a \cdot b^{t} \). The slope of the line in the semi-log plot relates to the growth factor over time.
Given the answer choice (C): \( N(t) = 2.5 \cdot 2^{(t/10)} \)
This means the population doubles every 10 days (since when \( t \) increases by 10, \( 2^{(t/10)} \) doubles). At \( t = 0 \), \( N(0) = 2.5 \) thousand, and at \( t = 10 \), \( N(10) = 2.5 \cdot 2 = 5 \) thousand, matching a point from the semi-log data if the plot is linear.
✅ Answer: (C)

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