Chapter 5: Exponential and Logarithmic Functions

Overview

• Focus: Exponential functions and their applications, including word problems.
• Later in Chapter 5: Introduction to logarithmic functions.
• Structure: Four sections covering exponential and logarithmic functions.

Exponential Functions

• Definition: Outputs are multiplied by a fixed number for each unit increase in the inputs.
• General Form:
  • ( f(x) = b^x ) or ( y = b^x )
  • ( b ) is the constant base, ( x ) is the variable exponent.

Types

1. Exponential Growth
   • Increases from left to right.
   • Graph: Crosses y-axis at (0,1), never crosses the x-axis (horizontal asymptote).
2. Exponential Decay
   • Decreases from left to right.
   • Graph: Similar to growth, with a horizontal asymptote along the x-axis.

Graphing with Calculators

• Method:
  1. Use graphing calculator (y = function, zoom 6).
  2. Determine if growth or decay by viewing graph.
  3. Example: ( y = 3^x + 2 ) (growth) and ( y = (\frac{1}{3})^x ) (decay).

Special Exponential Function: 'e'

• Constant 'e': Approx. 2.72, located on calculator.
• Example: ( y = e^x ) represents growth.

Calculations Using Exponential Functions

• Find specific function values: Use calculator table feature.
• Example Calculation: Find ( f(1), f(-4), f(4) ) using the table.

Word Problems with Exponential Models

• Example: Advertising and sales decay.
  • Formula: ( S = 1000 \times 2^{-0.5x} )
  • Calculate sales for specific weeks using tables and find intersection using graphs.
  • Horizontal Asymptote: Sales never reach exactly 0.

Investment Word Problems

• Annual Compounding Formula: ( S = P(1 + r)^t )

  • Calculate future investment value.

• Continuous Compounding Formula: ( S = Pe^{rt} )

  • Example: Doubling time for investments.
  • Use table and graph intersection to find specific years for investment growth.

Scientific Decay Problems

• Example: Radioactive isotope decay.

  • Formula: ( A = 800e^{-0.02852t} )
  • Calculate remaining amount after certain years.

• Graphical Estimation of Half-life: Use intersection method to find when half remains.

  • Example: Half-life calculated as 24.3 years.

Key Concepts

• Growth and Decay: Fundamental shapes and characteristics.
• Use of Calculators: For graphing, solving equations, and interpreting results.
• Word Problems: Application in sales, investments, and scientific decay.

Assignment

• Practice problems in MyMathLab covering exponential growth, decay, and word problems.