6.7 Exponential and Logarithmic Models

Exponential and Logarithmic Models

Exponential and logarithmic models describe how quantities grow, shrink, or level off over time. They show up constantly in real-world applications: population growth, radioactive decay, compound interest, and temperature change, to name a few. This section covers the major model types, when to use each one, and how to interpret their parameters.

Exponential and Logarithmic Models

Applications of exponential models

The two core exponential models differ by just a sign in the exponent, but they describe opposite behaviors.

Exponential growth: $A(t) = A_0 e^{kt}$

This models quantities that increase by a constant percentage over time.

• $A(t)$ = the quantity's value at time $t$
• $A_0$ = the initial value at $t = 0$
• $k$ = the growth rate (a positive constant)
• $t$ = time elapsed

Exponential decay: $A(t) = A_0 e^{-kt}$

This models quantities that decrease by a constant percentage over time. The variables mean the same thing, except $k$ now represents the decay rate. Notice the negative sign in the exponent is what makes the function decrease.

Common applications:

• Compound interest uses exponential growth. A bank account earning interest grows faster over time because you earn interest on your interest.
• Radioactive decay follows exponential decay. Carbon dating, for example, uses the known decay rate of carbon-14 to estimate the age of organic material.
• Population growth in ideal (unconstrained) conditions is exponential. Bacteria in a fresh nutrient solution double at regular intervals, which is classic exponential behavior.

Newton's Law of Cooling problems

Newton's Law of Cooling describes how an object's temperature changes as it moves toward the surrounding (ambient) temperature:

$T(t) = T_a + (T_0 - T_a)e^{-kt}$

• $T(t)$ = temperature of the object at time $t$
• $T_a$ = ambient temperature (the surrounding environment)
• $T_0$ = initial temperature of the object at $t = 0$
• $k$ = cooling rate constant (depends on the object and its environment)
• $t$ = time elapsed

The key idea: the difference between the object's temperature and the ambient temperature shrinks exponentially. A hot cup of coffee cools quickly at first (when the temperature gap is large), then more slowly as it approaches room temperature.

Typical problem-solving steps:

1. Identify the known values: $T_a$, $T_0$, and any temperature reading at a known time.
2. Plug the known temperature reading into the formula to solve for $k$.
3. Once you have $k$, use the formula to find the temperature at any other time, or solve for the time when the object reaches a target temperature.

For example, forensic investigators use this model to estimate time of death by measuring a body's current temperature, knowing the ambient temperature, and solving for $t$.

Logistic growth in populations

Pure exponential growth assumes unlimited resources, which isn't realistic. The logistic growth model accounts for limited resources by introducing a carrying capacity:

$P(t) = \frac{KP_0}{P_0 + (K - P_0)e^{-rt}}$

• $P(t)$ = population size at time $t$
• $P_0$ = initial population at $t = 0$
• $K$ = carrying capacity, the maximum population the environment can sustain
• $r$ = growth rate
• $t$ = time elapsed

The carrying capacity $K$ acts as a horizontal asymptote. The population can never exceed it (at least in the model).

How the curve behaves:

• Early on, when the population is small relative to $K$, growth looks nearly exponential.
• As the population grows and resources become scarcer, growth slows down.
• Eventually the population levels off near $K$.

Think of bacteria in a petri dish: they multiply rapidly at first, but as nutrients run low and waste builds up, growth tapers off. The S-shaped curve this produces is the signature of logistic growth.

Model selection for data sets

When you're given a data set and need to choose a model, look for these patterns:

1. Exponential growth: The quantity increases by a roughly constant percentage over equal time intervals. You might notice a consistent doubling time (e.g., the value doubles every 3 years).
2. Exponential decay: The quantity decreases by a roughly constant percentage over equal time intervals. Look for a consistent half-life (e.g., half the material remains every 5 hours).
3. Logistic growth: The data starts with rapid growth but then levels off toward a maximum value, producing an S-shaped curve.

Choosing and justifying your model:

• If growth appears unconstrained with no ceiling, an exponential model fits.
• If there's a clear upper limit the data approaches, a logistic model is more appropriate.
• Match the context to the model. Radioactive material decays exponentially. A population in a confined habitat grows logistically.

Once you've chosen a model type, you can use exponential regression or logarithmic regression (available on most graphing calculators) to fit the model to your data and find the best parameters.

Natural base e in exponentials

The number $e \approx 2.71828$ is a mathematical constant that appears naturally in growth and decay processes. You'll see it everywhere in these models, and there are good reasons for that.

Converting between bases:

Any exponential expression can be rewritten in base $e$:

$a^x = e^{x \ln a}$

Similarly, any logarithm can be converted to a natural logarithm:

$\log_a x = \frac{\ln x}{\ln a}$

These conversion formulas are useful when your calculator only has an $\ln$ button, or when you need to compare growth rates across models that use different bases.

Why base $e$ is preferred:

• Many natural processes (population growth, radioactive decay, cooling) are described most cleanly using base $e$.
• Continuously compounded interest uses the formula $A = A_0 e^{rt}$, which is the limiting case of compound interest as the compounding frequency increases without bound.
• In calculus, the derivative of $e^x$ is simply $e^x$, which makes base $e$ the most natural choice for mathematical analysis. Even in this algebra course, using $e$ keeps formulas cleaner.

Rate of change and logarithmic scale

Exponential and logarithmic functions have distinctive rate-of-change behavior:

• Exponential models have a rate of change that's proportional to the current value. The larger the quantity, the faster it grows (or decays). That's why exponential growth feels slow at first and then seems to explode.
• Logarithmic models have a rate of change that's inversely proportional to the input. Large inputs produce smaller and smaller changes. This is why logarithmic growth flattens out over time.

Logarithmic scales take advantage of this property to display data that spans many orders of magnitude. Instead of spacing values evenly (1, 2, 3, 4...), a log scale spaces them by powers of 10 (1, 10, 100, 1000...). The Richter scale for earthquake intensity and the decibel scale for sound intensity both use logarithmic scales. On these scales, each step up represents a tenfold increase in the actual quantity, which makes it possible to compare very small and very large values on the same graph.