8.4 Applications of Exponential and Logarithmic Functions

Exponential and logarithmic functions model situations involving growth, decay, and scaling. They show up constantly in biology, finance, chemistry, and physics, so learning to apply them is just as important as learning to manipulate them algebraically.

This section covers how to set up and solve real-world problems using exponential and logarithmic models, and how to interpret what your answers actually mean.

Modeling Real-World Situations

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Exponential Functions for Growth and Decay

Exponential functions model situations where a quantity changes by a constant percentage (not a constant amount) over equal time intervals. That distinction matters: adding 50 people per year is linear, but growing by 5% per year is exponential.

The general form is:

$f(x) = a \cdot b^x$

• $a$ = the initial value (the amount when $x = 0$)
• $b$ = the growth or decay factor

If $b > 1$, the function models growth. If $0 < b < 1$, it models decay.

• Growth examples: population increase, compound interest, viral spread
• Decay examples: radioactive decay, depreciation of a car's value, cooling of a hot object

Logarithmic Functions as Inverses

Logarithmic functions reverse what exponential functions do. If an exponential function answers "what do I get after this many time periods?", a logarithm answers "how many time periods until I reach this value?"

The general form is:

$f(x) = \log_b(x)$

where $b$ is the base of the logarithm.

Logarithmic scales compress huge ranges of values into manageable numbers. Three important ones to know:

• Richter scale (earthquakes): each whole number increase represents a tenfold increase in measured amplitude
• pH scale (acidity): each unit decrease in pH means a tenfold increase in hydrogen ion concentration
• Decibel scale (sound): each increase of 10 dB represents a tenfold increase in sound intensity

Exponential Growth and Decay

Formulas and Key Concepts

Exponential growth (continuous model):

$A(t) = A_0 \cdot e^{kt}$

• $A_0$ = initial amount
• $k$ = growth rate (a positive constant)
• $t$ = time elapsed

Doubling time is how long it takes the quantity to double:

$t_{\text{double}} = \frac{\ln(2)}{k}$

Exponential decay uses the same structure but with a negative exponent:

$A(t) = A_0 \cdot e^{-kt}$

Half-life is how long it takes the quantity to drop to half its value:

$t_{1/2} = \frac{\ln(2)}{k}$

Notice that the doubling time and half-life formulas look identical. The difference is just context: one applies to growing quantities, the other to decaying ones.

Solving Problems with Exponential Models

Here's a reliable process for these problems:

1. Identify your variables. What's the initial amount $A_0$? What's the rate $k$? What's the time $t$? Which variable are you solving for?
2. Choose the right formula. Use the growth formula (positive $k$) or decay formula (negative exponent) depending on the situation.
3. Substitute known values into the formula.
4. Solve for the unknown. If the unknown is in the exponent, you'll need to take a logarithm of both sides.

Example: A bacteria colony starts with 500 cells and grows continuously at a rate of $k = 0.12$ per hour. How many cells are there after 10 hours?

$A(10) = 500 \cdot e^{0.12 \cdot 10} = 500 \cdot e^{1.2} \approx 500 \cdot 3.32 \approx 1660 \text{ cells}$

Example (solving for time): How long until the colony reaches 5000 cells?

$5000 = 500 \cdot e^{0.12t}$ $10 = e^{0.12t}$ $\ln(10) = 0.12t$ $t = \frac{\ln(10)}{0.12} \approx \frac{2.303}{0.12} \approx 19.2 \text{ hours}$

Logarithms in Applications

Chemistry and Physics

pH (Chemistry):

$\text{pH} = -\log[H^+]$

where $[H^+]$ is the hydrogen ion concentration in moles per liter.

A solution with $[H^+] = 10^{-5}$ mol/L has a pH of 5. A solution with $[H^+] = 10^{-2}$ mol/L has a pH of 2. The lower the pH, the more acidic the solution. Each one-unit drop in pH means the hydrogen ion concentration is 10 times greater.

Decibels (Physics):

$L = 10 \cdot \log\left(\frac{I}{I_0}\right)$

where $I$ is the sound intensity and $I_0$ is the reference intensity ($10^{-12}$ W/m²). A normal conversation is about 60 dB, while a rock concert might hit 110 dB. That 50 dB difference means the concert is $10^5 = 100{,}000$ times more intense, not just "a little louder."

Finance and Other Fields

Continuously compounded interest:

$A(t) = P \cdot e^{rt}$

• $P$ = principal (starting amount)
• $r$ = annual interest rate (as a decimal)
• $t$ = time in years

Example: You invest $2,000 at 4.5% interest compounded continuously. What's it worth after 8 years?

$A(8) = 2000 \cdot e^{0.045 \cdot 8} = 2000 \cdot e^{0.36} \approx 2000 \cdot 1.4333 \approx \$2{,}866.63$

Finding time to reach a target: If you need to know when an investment hits a certain value, isolate $t$ by dividing both sides by $P$, then taking $\ln$ of both sides. This is where your logarithm properties (product rule, quotient rule, power rule) become essential tools, not just abstract rules.

Interpreting Model Results

Understanding the Context

Don't just calculate a number; make sure you understand what it means.

For exponential models, pay attention to the growth/decay factor. If a population model has $b = 1.05$, that means a 5% increase each year. If a car's value has $b = 0.82$, it loses 18% of its value annually.

For logarithmic models, think about what the input and output represent. A pH of 3 means $[H^+] = 10^{-3}$ mol/L. Going from pH 3 to pH 1 doesn't mean "a little more acidic"; it means the hydrogen ion concentration increased by a factor of 100.

Limitations and Assumptions

Every model simplifies reality. Recognizing those simplifications is part of using models responsibly.

• Exponential growth models assume unlimited resources. A bacteria population can't grow exponentially forever; eventually food runs out or space becomes limited. In ecology, this is why logistic models eventually replace exponential ones.
• Continuous compounding assumes a fixed rate. Real interest rates fluctuate, and investments carry risk.
• Logarithmic scales compress extremes. A magnitude 7.0 earthquake isn't "a little worse" than a 6.0; it releases roughly 31.6 times more energy.

When you present results from a model, state your assumptions clearly. If you predict a city's population will reach 2 million in 15 years using an exponential model, acknowledge that the growth rate could change due to migration, policy, or economic shifts.