Exponential and Logarithmic Functions: Applications in Business Calculus

Chapter 2: Exponential and Logarithmic Functions

Introduction

This chapter explores the mathematical properties and real-world applications of exponential and logarithmic functions, with a focus on exponential decay. These concepts are foundational in business calculus and are widely used in fields such as finance, physical sciences, and life sciences.

Applications: Exponential Decay

Objectives

• Define a function that satisfies $\frac{dP}{dt} = -kP$.

• Convert between decay rate and half-life.

• Solve applied problems involving exponential decay.

Exponential Decay Equation

Exponential decay describes processes where a quantity decreases at a rate proportional to its current value. The general form of the differential equation is:

• $\frac{dP}{dt} = -kP$, where $k > 0$

• The solution to this equation is $P(t) = P_0 e^{-kt}$

• $P_0$ is the initial amount at time $t = 0$

This equation models exponential decrease, such as radioactive decay, depreciation, and cooling.

Exponential Growth vs. Decay

• Growth: $P = P_0 e^{kt}$ for $k > 0$

• Decay: $P = P_0 e^{-kt}$ for $k > 0$

Growth curves rise over time, while decay curves fall, both starting from $P_0$.

Theorem 10: Relationship Between Decay Rate and Half-Life

The decay rate $k$ and the half-life $T$ are related by:

• $T = \frac{\ln 2}{k}$

• $k = \frac{\ln 2}{T}$

Half-life is the time required for a quantity to reduce to half its initial value.

Example 2: Physical Science – Half-life Calculation

Plutonium-239, a radioactive product, has a decay rate of 0.0028% per year. Its half-life is calculated as:

• $T = \frac{\ln 2}{0.000028} \approx 24,755$ years

Thus, the half-life of plutonium-239 is about 24,755 years.

Check 2: Practice Problems

• Cesium-137: Decay rate $k = 0.023$ per year

• $T = \frac{\ln 2}{0.023} \approx 30.1$ years

• Barium-140: Half-life $T = 13$ days

• $k = \frac{\ln 2}{13} \approx 0.053$ or $5.3\%$ per day

Example 3: Life Science – Carbon Dating

Carbon-14 has a half-life of 5730 years. The percentage of carbon-14 in remains is used to determine age. For the Dead Sea Scrolls, which lost 22.3% of carbon-14:

• Decay rate: $k = \frac{\ln 2}{5730} \approx 0.00012097$ or $0.012097\%$ per year

Example 3 Continued: Calculating Age Using Decay

• Let $N = N_0 e^{-kt}$

• Given $N/N_0 = 0.777$ (77.7% remains)

• $0.777 = e^{-0.00012097 t}$

• $\ln 0.777 = -0.00012097 t$

• $t = \frac{\ln 0.777}{-0.00012097} \approx 2086$ years

The linen wrapping from the Dead Sea Scroll is about 2086 years old.

Check 3: Practice – Carbon Dating

• If a skeleton has lost 60% of its carbon-14, then $N/N_0 = 0.4$

• $0.4 = e^{-0.00012097 t}$

• $t = \frac{\ln 0.4}{-0.00012097} \approx 7574$ years

The skeleton is approximately 7574 years old.

Summary Table: Decay Rate and Half-Life Calculations

Isotope	Decay Rate ($k$)	Half-Life ($T$)	Calculation
Plutonium-239	0.000028 per year	24,755 years	$T = \frac{\ln 2}{0.000028}$
Cesium-137	0.023 per year	30.1 years	$T = \frac{\ln 2}{0.023}$
Barium-140	0.053 per day	13 days	$k = \frac{\ln 2}{13}$
Carbon-14	0.00012097 per year	5730 years	$k = \frac{\ln 2}{5730}$

Key Terms and Concepts

• Exponential Decay: A process where a quantity decreases at a rate proportional to its current value.

• Decay Rate ($k$): The proportionality constant in the decay equation.

• Half-Life ($T$): The time required for a substance to decrease to half its initial amount.

• Carbon Dating: A method for determining the age of an object containing organic material by measuring the amount of carbon-14.

Applications in Business Calculus

• Exponential decay models are used in finance (e.g., depreciation), science (e.g., radioactive decay), and forensics (e.g., time of death estimation).

• Understanding the relationship between decay rate and half-life is essential for interpreting real-world data and making predictions.