2.14 Logarithmic function context and data modeling
Given the following exponential functions, find their equation of the logarithmic functions that serve as their inverse.
f(x) = 2^x
g(x) = 3^{x - 1}
h(x) = 5^{2x - 3}
p(x) = 7^{3x + 2}
Given the following logarithmic functions, describe the transformations needed to obtain the corresponding functions.
f(x) = \log_2(x + 1) \to g(x) = -2\log_2(-x) + 3
g(x) = \log_3(x) \to h(x) = \log_3(x - 4) + 2
p(x) = \log_4(x) \to q(x) = -2\log_4(x - 3)
a(x) = \log_5(x) \to b(x) = \log_5(2x + 1) - 3
Given the function f(x) = 2 \log(x), describe how the function model can be transformed to represent the situation where the growth factor is doubled and the initial value is shifted 3 units upwards.
The population of a town grows at a proportional rate of 8\% per year. Write a logarithmic function model that represents the population growth of the town.
The number of users of a certain mobile application increases at a proportional rate of 15\% per year. Write a logarithmic function model that represents the user growth of the mobile application.
Given the proportion 2:5 and the real zero 3, construct a logarithmic function model that represents this situation.
Given the following input-output pairs, construct a logarithmic function model that fits each data points.
(1, 0) and (10, 1)
(2, 0) and (4, 1)
(3, 0) and (9, 1)
(4, 0) and (16, 2)
Use a calculator or any form of technology to perform a logarithmic regression on the following data sets and construct a logarithmic function that best fits each data:
{(1, 2), (2, 4), (3, 8), (4, 16) (5, 32)}
{(1, 0.3), (2, 0.9), (3, 1.4), (4, 1.8), (5, 2.1)}
Given the logarithmic function model f(x) = \log_2(x - 3).
Predict the output values for x = 4, x = 5, and x = 6.
Interpret the meaning of these predictions.
A city's population, P, has been growing logarithmically. In 1990, the population was 5000, and by 2010 it had grown to 20000. Construct a logarithmic model for the population of the city over time.
The sound level L in decibels of a sound is given by L = 10 \log_{10}\dfrac{\text{I}}{\text{I}_0}, where I is the sound's intensity and \text{I}_0 is a reference intensity. If a sound's intensity is 1000 times the reference intensity, what is its sound level?
The acidity of a solution is measured by its pH level, which is a logarithmic measure. If a solution has 10 times the hydrogen ion concentration of a neutral solution (pH 7), what is its pH?
A biologist is studying the growth of a species of bacteria. The bacteria count is found to double every hour. After 5 hours, the bacteria count was 1200.
Write a logarithmic function model that describes this situation.
Use your model to predict the bacteria count after 8 hours.
In a physics experiment, the light intensity is measured by a sensor and represented by the function \text{I}(x) = \log(x), where x is the light source intensity. If the light source intensity doubles, what is the new function representing the changed light intensity?
Astronomers measure the brightness of stars using a logarithmic function, B(x) = -2.5\log(x), where x is the star's brightness. If the star's brightness increases tenfold, what is the new function representing the changed brightness?
Explain why the natural logarithm function (\ln x) is often used in modeling real-world phenomena. Provide an example.
When is a logarithmic function model useful over an exponential function model in a real-world context?
The intensity of an earthquake is typically measured using a logarithmic scale called the Richter scale. An earthquake that measures 5.0 on the Richter scale is 10 times more intense than an earthquake that measures4.0.
Explain why a logarithmic scale might be a more appropriate way to measure earthquake intensity than a linear scale.
If an earthquake measures 7.2 on the Richter scale, how many times more intense is it than an earthquake that measures 4.5?
A car's value depreciates over time, and the rate of depreciation is proportional to the car's current value. The car is initially worth \$20,000, and after 5 years it's worth \$13,000.
Construct a logarithmic model for the car's value over time where lettingk as constant.
Find the value of k.
An economist is using the natural logarithm function to model economic growth over time.
Explain why the natural logarithm function might be a useful tool for this kind of modeling.
Give of any potential limitations or challenges of using a logarithmic function model in this context?