Exponential and Logarithmic Models: Fresh Take

  • Model situations of increasing or decreasing values, like population growth or radioactive decay, and use special equations like Newton’s Law for cooling
  • Understand how to use logistic models for growth that slows down as it reaches a limit
  • Figure out when to use exponential models to analyze real-world data

Exponential Growth and Decay

The Main Idea

  • Exponential Growth: [latex]A = A_0e^{rt}[/latex]
    • [latex]A_0[/latex]: Initial amount
    • [latex]r[/latex]: Growth rate (positive)
    • [latex]t[/latex]: Time
  • Exponential Decay: [latex]A = A_0e^{-kt}[/latex]
    • [latex]k[/latex]: Decay rate (positive)
  • Doubling Time: [latex]t = \frac{\ln(2)}{r}[/latex]
  • Half-life: [latex]t = \frac{\ln(2)}{k} = \frac{\ln(1/2)}{-k}[/latex]

Characteristics of Exponential Functions

  • Domain: All real numbers
  • Range: [latex](0, ∞)[/latex]
  • y-intercept: [latex](0, A_0)[/latex]
  • Horizontal asymptote: [latex]y = 0[/latex]
  • Increasing if [latex]r > 0[/latex], decreasing if [latex]r < 0[/latex]

Applications

  1. Population Growth
  2. Compound Interest
  3. Moore’s Law (computing power)
  4. Radioactive Decay
  5. Radiocarbon Dating

Problem-Solving Approach

  1. Identify whether it’s growth or decay
  2. Determine the initial amount ([latex]A_0[/latex])
  3. Calculate or identify the rate ([latex]r[/latex] or [latex]k[/latex])
  4. Apply the appropriate formula
  5. Solve for the unknown variable (often time [latex]t[/latex] or final amount [latex]A[/latex])

Key Formulas for Radioactive Decay

  1. General form: [latex]A(t) = A_0(\frac{1}{2})^{t/T}[/latex] where [latex]T[/latex] is the half-life
  2. Radiocarbon dating: [latex]t = \frac{\ln(A/A_0)}{-0.000121}[/latex] where [latex]A/A_0[/latex] is the ratio of current to initial carbon-14
Recent data suggests that, as of 2013, the rate of growth predicted by Moore’s Law no longer holds. Growth has slowed to a doubling time of approximately three years. Find the new function that takes that longer doubling time into account.

Uranium-235 has a half-life of [latex]703,800,000[/latex] years. How long will it take before twenty percent of our [latex]1000[/latex]-gram sample of uranium-235 has decayed?

The half-life of plutonium-244 is [latex]80,000,000[/latex] years. Find a function that gives the amount of plutonium-244 remaining as a function of time measured in years.

Cesium-137 has a half-life of about 30 years. If we begin with 200 mg of cesium-137, will it take more or less than 230 years until only 1 milligram remains?

You can view the transcript for “Exponential Growth App (y=ab^t) – Find Initial Amount Given Doubling Time” here (opens in new window).

You can view the transcript for “Exponential Growth App (y=ab^t) – Given Doubling Time” here (opens in new window).

You can view the transcript for “Ex: Exponential Model – Determine Age Using Carbon-14 Given Half Life” here (opens in new window).

Newton’s Law of Cooling

The Main Idea

Newton’s Law of Cooling describes how an object’s temperature changes exponentially as it approaches the ambient temperature:

[latex]T(t) = A e^{kt} + T_s[/latex]

Where:

  • [latex]T(t)[/latex] is the temperature at time [latex]t[/latex]
  • [latex]A[/latex] is the initial temperature difference (object – surroundings)
  • [latex]k[/latex] is the cooling rate (negative for cooling)
  • [latex]T_s[/latex] is the surrounding temperature

Characteristics

  1. Exponential decay towards ambient temperature
  2. Vertical shift of the exponential decay function
  3. Horizontal asymptote at [latex]T_s[/latex]

Problem-Solving Approach

  1. Identify [latex]T_s[/latex] (ambient temperature)
  2. Find [latex]A[/latex] by subtracting [latex]T_s[/latex] from initial temperature
  3. Use a second data point to solve for [latex]k[/latex]
  4. Apply the formula to find unknown time or temperature
A pitcher of water at 40 degrees Fahrenheit is placed into a 70 degree room. One hour later, the temperature has risen to 45 degrees. How long will it take for the temperature to rise to 60 degrees?

You can view the transcript for “Ex: Newton’s Law of Cooling – Exponential Function App” here (opens in new window).

Logistic Growth

The Main Idea

The logistic growth model describes growth with a limiting factor:

[latex]f(x) = \frac{c}{1 + ae^{-bx}}[/latex]

Where:

  • [latex]c[/latex] is the carrying capacity (upper limit)
  • [latex]a[/latex] affects the initial value (f(0) = c/(1+a))
  • [latex]b[/latex] is the growth rate

Characteristics

  1. S-shaped curve
  2. Initially similar to exponential growth
  3. Growth rate decreases as it approaches carrying capacity
  4. Horizontal asymptote at [latex]y = c[/latex]

Comparison to Exponential Growth

  1. Exponential: Unlimited growth
  2. Logistic: Limited by carrying capacity
  3. Exponential: Constant relative growth rate
  4. Logistic: Decreasing relative growth rate

Applications

  1. Population growth with limited resources
  2. Spread of infectious diseases
  3. Technology adoption
  4. Product sales over time
An influenza epidemic spreads through a population rapidly at a rate that depends on two factors. The more people who have the flu, the more rapidly it spreads, and also the more uninfected people there are, the more rapidly it spreads. These two factors make the logistic model good for studying the spread of communicable diseases. And, clearly, there is a maximum value for the number of people infected: the entire population.For example, at time [latex]t = 0[/latex] there is one person in a community of [latex]1,000[/latex] people who has the flu. So, in that community, at most [latex]1,000[/latex] people can have the flu. Researchers find that for this particular strain of the flu, the logistic growth constant is [latex]b = 0.6030[/latex]. Estimate the number of cases of flu on day [latex]15[/latex].

Exponential Regression

The Main Idea

Exponential regression is used to model data that shows exponential growth or decay. The general form of the model is:

[latex]y = ab^x[/latex]

Where:

  • [latex]a[/latex] is the [latex]y[/latex]-intercept (initial value)
  • [latex]b[/latex] is the base (growth/decay factor)

Characteristics

  1. For growth: [latex]b > 1[/latex]
  2. For decay: [latex]0 < b < 1[/latex]
  3. [latex]a[/latex] must be positive
  4. Initial growth/decay is slow, then accelerates

When to Use Exponential Regression

  • Data increases/decreases by a constant percentage
  • Growth starts slow, then accelerates rapidly
  • Decay is rapid at first, then slows down approaching zero

Regression Process

  1. Enter data into a graphing utility
  2. Create a scatter plot
  3. Perform exponential regression
  4. Analyze the fit (r² value)
  5. Graph the model with the data points
The table below shows a recent graduate’s credit card balance each month after graduation.

Month 1 2 3 4 5 6 7 8
Debt ($) 620.00 761.88 899.80 1039.93 1270.63 1589.04 1851.31 2154.92
  1. Use exponential regression to fit a model to these data.
  2. If spending continues at this rate, what will the graduate’s credit card debt be one year after graduating?

You can view the transcript for “Ex: Perform Exponential Regression on a Graphing Calculator” here (opens in new window).