Logarithmic Function Context and Data Modeling
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AP Precalculus › Logarithmic Function Context and Data Modeling
The intensity of sound, measured in decibels, can be modeled by $$I(d) = 85 - 12\log_{10}(d)$$, where $$d$$ is the distance in meters from the sound source. According to this model, what happens to the sound intensity when the distance from the source increases from 10 meters to 100 meters?
The sound intensity decreases by exactly 12 decibels due to the logarithmic relationship between distance and intensity.
The sound intensity remains constant at 85 decibels since the logarithmic term becomes negligible at large distances.
The sound intensity increases by exactly 12 decibels because the logarithmic function amplifies distance effects.
The sound intensity decreases by exactly 24 decibels because the distance increases by a factor of 10.
Explanation
When distance increases from 10 to 100 meters, we calculate: $$I(10) = 85 - 12\log_{10}(10) = 85 - 12(1) = 73$$ and $$I(100) = 85 - 12\log_{10}(100) = 85 - 12(2) = 61$$. The intensity decreases from 73 to 61 decibels, a decrease of 12 decibels. This occurs because $$\log_{10}(100) - \log_{10}(10) = 2 - 1 = 1$$, and the coefficient -12 means the intensity decreases by 12 decibels.
A psychology experiment measures reaction time in milliseconds using the model $$R(n) = 180 + 25\ln(n + 2)$$, where $$n$$ is the number of previous trials completed. Which statement best describes what this model predicts about reaction times?
Reaction times will oscillate around the baseline value of 180 milliseconds, with the amplitude of oscillation determined by the natural logarithm term.
Reaction times will decrease rapidly at first, then level off as participants become more experienced with the experimental task over time.
Reaction times will continue to increase indefinitely as the number of trials increases, following a linear growth pattern throughout the experiment.
Reaction times will increase initially but at a decreasing rate, eventually approaching a maximum value as participants reach cognitive saturation.
Explanation
Since the coefficient of $$\ln(n + 2)$$ is positive (25), reaction times increase as $$n$$ increases. However, the logarithmic function increases at a decreasing rate - it grows quickly at first, then more slowly. This means reaction times increase initially but the rate of increase slows down, which could represent cognitive fatigue or task saturation effects in psychology experiments.
A seismologist uses the model $$M(E) = 2.3\log_{10}(E) - 5.8$$ to relate earthquake magnitude $$M$$ to energy released $$E$$ (in joules). If an earthquake releases $$10^8$$ joules of energy, what is its magnitude on this scale?
7.8 on the magnitude scale, found by evaluating the logarithmic function at the specified energy level.
18.4 on the magnitude scale, calculated using the base-10 logarithm and given scaling coefficients.
12.6 on the magnitude scale, computed by substituting the energy value and evaluating the logarithmic expression.
2.6 on the magnitude scale, determined by applying logarithmic properties to the energy measurement.
Explanation
Substituting $$E = 10^8$$: $$M(10^8) = 2.3\log_{10}(10^8) - 5.8 = 2.3(8) - 5.8 = 18.4 - 5.8 = 12.6$$. The magnitude is 12.6 on this scale. Note that $$\log_{10}(10^8) = 8$$ by the definition of logarithms.
A pharmacologist models drug concentration in blood plasma using $$C(t) = 50 - 12\ln(t + 1)$$, where $$t$$ is time in hours after administration. According to this model, how does the concentration change between $$t = 0$$ and $$t = 2$$ hours?
The concentration decreases by approximately 24.0 mg/L due to rapid drug metabolism and elimination processes modeled by the logarithmic function.
The concentration decreases by approximately 13.2 mg/L as the logarithmic decay model predicts drug elimination from the bloodstream over time.
The concentration remains essentially constant at 50 mg/L since the logarithmic term has minimal impact during the first few hours.
The concentration increases by approximately 8.7 mg/L because the natural logarithm coefficient amplifies the drug absorption rate during this period.
Explanation
At $$t = 0$$: $$C(0) = 50 - 12\ln(0 + 1) = 50 - 12\ln(1) = 50 - 12(0) = 50$$ mg/L. At $$t = 2$$: $$C(2) = 50 - 12\ln(2 + 1) = 50 - 12\ln(3) = 50 - 12(1.099) = 50 - 13.19 = 36.8$$ mg/L. The concentration decreases from 50 to 36.8 mg/L, a decrease of approximately 13.2 mg/L.
An economist models inflation impact using $$I(t) = 3.2 + 1.5\ln(t + 3)$$, where $$t$$ is years since policy implementation and $$I(t)$$ is the inflation rate percentage. What was the inflation rate 4 years before policy implementation?
The inflation rate was 1.56% since the logarithmic model can be extrapolated backward to negative time values using standard methods.
This question cannot be answered because $$t = -4$$ makes the logarithm argument negative, placing it outside the model's valid domain.
The inflation rate was 0.00% because the logarithmic function approaches zero as the time parameter becomes increasingly negative.
The inflation rate was 4.85% calculated by substituting $$t = -4$$ into the economic model and evaluating the logarithmic expression.
Explanation
To find the inflation rate 4 years before policy implementation, we need $$t = -4$$. This gives us $$I(-4) = 3.2 + 1.5\ln(-4 + 3) = 3.2 + 1.5\ln(-1)$$. Since the natural logarithm is undefined for negative arguments, $$t = -4$$ is outside the domain of this model. The model is only valid for $$t > -3$$ years.
A chemistry class measures a radioactive sample that steadily loses mass. The instructor emphasizes that the process is exponential with a constant decay rate, so a log transformation linearizes the data. They model mass by $m(t)=m_0(1-d)^t$ and rewrite it as $\log_{10}(m)=\log_{10}(m_0)+t,\log_{10}(1-d)$. Here the logarithm uses base 10, and the slope equals $\log_{10}(1-d)$. The table gives measured masses in grams at integer hours. Students are asked to pick the log-linear equation that matches the measurements.
Based on the scenario, which logarithmic equation represents the scenario described?
$\log_{10}(m)=\log_{10}(45)+t,\log_{10}(0.95)$
$m=50,\log_{10}(0.95t)$
$\log_{10}(m)=\log_{10}(50)+t,\log_{10}(0.90)$
$\log_{10}(m)=\log_{10}(50)+t,\log_{10}(0.95)$
Explanation
This question tests AP Precalculus understanding of logarithmic functions and data modeling, specifically applying logarithms to radioactive decay with given initial conditions and decay rates. The exponential decay model m(t) = $m₀(1-d)^t$ becomes linear when log-transformed, allowing for easier analysis and prediction. In this scenario, the initial mass is m₀ = 50 grams, and the decay results in retaining 95% of mass each hour (decay rate d = 0.05, so 1-d = 0.95). Choice A is correct because it accurately represents the logarithmic form: log₁₀(m) = log₁₀(m₀) + t·log₁₀(1-d) = log₁₀(50) + t·log₁₀(0.95). Choice C is incorrect because it uses m₀ = 45 instead of 50, misreading the initial mass from the table. To help students: Emphasize careful reading of data tables to identify initial values (at t=0) versus later measurements. Practice setting up logarithmic models by first writing the exponential form, then applying logarithms to both sides systematically.
A computer scientist models processing time using $$T(n) = 0.5 + 0.12\log_2(n)$$, where $$n$$ is the input size and $$T(n)$$ is time in seconds. If the processing time is 1.94 seconds, what is the input size?
12288 units, obtained by substituting the known processing time and solving the resulting base-2 logarithmic equation through standard methods.
4096 units, calculated by solving the logarithmic equation and converting to exponential form using base-2 properties and algebraic manipulation.
8192 units, determined by isolating the logarithmic term and applying inverse operations to find the input size systematically.
16384 units, found by setting up the time equation and using logarithmic properties to solve for the computational input parameter.
Explanation
Setting $$T(n) = 1.94$$: $$1.94 = 0.5 + 0.12\log_2(n)$$. Subtracting 0.5: $$1.44 = 0.12\log_2(n)$$. Dividing by 0.12: $$12 = \log_2(n)$$. Converting to exponential form: $$n = 2^{12} = 4096$$ units.
A sociologist studying urban population density uses the model $$D(r) = 2500 - 180\log_3(r + 2)$$, where $$r$$ is the distance in kilometers from the city center. At what distance from the center does the model predict a density of 2140 people per square kilometer?
43 kilometers from the city center, obtained by isolating the logarithmic term and converting to exponential form using standard algebraic methods.
16 kilometers from the city center, determined by substituting the target density and solving the resulting base-3 logarithmic equation systematically.
25 kilometers from the city center, found by setting up the density equation and using logarithmic properties to determine the radial distance.
7 kilometers from the city center, calculated by solving the logarithmic equation and applying the inverse operations to isolate the distance variable.
Explanation
Setting $$D(r) = 2140$$: $$2140 = 2500 - 180\log_3(r + 2)$$. Subtracting 2500: $$-360 = -180\log_3(r + 2)$$. Dividing by -180: $$2 = \log_3(r + 2)$$. Converting to exponential form: $$3^2 = r + 2$$, so $$9 = r + 2$$. Therefore, $$r = 7$$ kilometers from the city center.
A financial analyst models the relationship between company size and market share using $$S(n) = 15 + 8\log_2(n - 5)$$, where $$n$$ is the number of employees (in hundreds) and $$S(n)$$ is market share percentage. What is the domain restriction for this model in the business context?
$$n \geq 15$$ hundred employees, since the market share percentage cannot be negative and companies need minimum staff for market participation.
$$n \geq 8$$ hundred employees, as the logarithmic scaling factor requires sufficient company size to produce meaningful market share calculations.
$$n > 0$$ hundred employees, because companies must have at least some employees to operate and generate any measurable market share.
$$n > 5$$ hundred employees, because the logarithmic argument must be positive and represents the effective workforce size above a threshold.
Explanation
For the logarithmic function $$\log_2(n - 5)$$ to be defined, the argument $$(n - 5)$$ must be positive. This means $$n - 5 > 0$$, so $$n > 5$$. In the business context, this suggests that companies need more than 500 employees (5 hundred) before this market share model becomes applicable, possibly representing a threshold above which the logarithmic relationship holds.
A marine biologist models fish population recovery using $$N(m) = 1200 + 340\log_5(m + 6)$$, where $$m$$ is months after conservation efforts began. How many additional months are needed for the population to increase from 2000 to 2200 fish?
Approximately 8.4 additional months, calculated by using the logarithmic model properties and the specified population increase requirements.
Approximately 19.2 additional months, found by solving two logarithmic equations and calculating the difference between the corresponding time values.
Approximately 47.3 additional months, obtained by solving the conservation model equations and applying the population growth time differential.
Approximately 31.7 additional months, determined by setting up equations for both population levels and finding the time interval between them.
Explanation
First, find when $$N(m) = 2000$$: $$2000 = 1200 + 340\log_5(m + 6)$$, so $$800 = 340\log_5(m + 6)$$, giving $$\log_5(m + 6) = \frac{800}{340} \approx 2.35$$. Thus $$m + 6 = 5^{2.35} \approx 19.2$$, so $$m \approx 13.2$$ months. Next, find when $$N(m) = 2200$$: $$2200 = 1200 + 340\log_5(m + 6)$$, so $$1000 = 340\log_5(m + 6)$$, giving $$\log_5(m + 6) = \frac{1000}{340} \approx 2.94$$. Thus $$m + 6 = 5^{2.94} \approx 32.4$$, so $$m \approx 26.4$$ months. The additional time needed is $$32.4 - 13.2 = 19.2$$ months.