Exponential and Logarithmic Models

Overview

Exponential and logarithmic models are powerful tools used to describe real-world phenomena such as population growth, radioactive decay, and sound intensity. These models rely on exponential and logarithmic functions to represent relationships between variables.

Two common forms of these models are:

  • Exponential Growth/Decay: \( y = A e^{kt} \)
  • Logarithmic Models: \( y = A \ln(x) + B \)

Key Concepts

1. Exponential Growth and Decay

Exponential growth occurs when a quantity increases at a constant percentage rate, modeled by \( y = A e^{kt} \), where:

  • \( y \): final amount
  • \( A \): initial amount
  • \( k \): growth (\( k > 0 \)) or decay (\( k < 0 \)) constant
  • \( t \): time

Example: A population of 1000 grows at a rate of 5% per year. Find the population after 10 years.

Solution: Use \( y = A e^{kt} \)

Step 1: \( y = 1000 \cdot e^{0.05 \cdot 10} \)

Step 2: \( y \approx 1648.72 \)

2. Logarithmic Models

Logarithmic models are often used to describe diminishing returns or processes that slow down over time, such as the pH scale or sound intensity.

Example: The brightness of a star decreases logarithmically as the distance from it increases. The model is \( y = 10 \ln(x) + 50 \), where \( y \) is brightness, and \( x \) is distance. Find the brightness when \( x = 2 \).

Solution: \( y = 10 \ln(2) + 50 \)

Step 1: \( y \approx 10 \cdot 0.693 + 50 \)

Step 2: \( y \approx 56.93 \)

Practice Questions

  1. Question 1: A radioactive substance has a half-life of 10 years. The initial amount is 200 grams. Write the model and find the remaining amount after 30 years.
    Solution

    Model: \( y = A e^{kt} \), where \( k = \ln(0.5)/10 \)

    Step 1: \( y = 200 \cdot e^{\ln(0.5) \cdot 3} \)

    Step 2: \( y \approx 25 \) grams

  2. Question 2: The sound intensity level in decibels (dB) is modeled as \( L = 10 \log(I/I_0) \), where \( I_0 \) is a reference intensity. If \( I = 100I_0 \), find \( L \).
    Solution

    Step 1: Substitute into the model: \( L = 10 \log(100) \)

    Step 2: \( L = 10 \cdot 2 \)

    Step 3: \( L = 20 \) dB

  3. Question 3: A population of bacteria doubles every 3 hours. If the initial population is 500, find the population after 9 hours.
    Solution

    Step 1: Use \( y = A e^{kt} \), where \( k = \ln(2)/3 \)

    Step 2: \( y = 500 \cdot e^{(\ln(2)/3) \cdot 9} \)

    Step 3: \( y \approx 4000 \)

  4. Question 4: The growth of a city's population follows the model \( P(t) = 1000e^{0.02t} \). Find the time \( t \) when the population reaches 1500.
    Solution

    Step 1: Solve \( 1500 = 1000e^{0.02t} \)

    Step 2: \( 1.5 = e^{0.02t} \)

    Step 3: Take the natural logarithm: \( \ln(1.5) = 0.02t \)

    Step 4: Solve for \( t \): \( t \approx 20.7 \) years

  5. Question 5: The brightness of a bulb decreases logarithmically with time \( t \) as \( B(t) = 50 - 5 \ln(t) \). Find the brightness at \( t = 10 \).
    Solution

    Step 1: \( B(10) = 50 - 5 \ln(10) \)

    Step 2: \( B(10) = 50 - 5 \cdot 2.302 \)

    Step 3: \( B(10) \approx 38.49 \)

Extra Practice

  • A car's value decreases exponentially at a rate of 15% per year. If its initial value is $20,000, find its value after 5 years.
  • The pH of a solution is modeled as \( \text{pH} = -\log[H^+] \). If \( [H^+] = 1 \times 10^{-5} \), find the pH.
  • Write a model for the decay of 100 grams of a substance with a half-life of 12 hours.
  • Find the time it takes for an investment to double at an annual interest rate of 7% compounded continuously.
  • The population of a town follows the model \( P(t) = 5000e^{0.03t} \). Find the population after 8 years.
→Transformations of Exponential and Logarithmic Functions