Exponential and Logarithmic Functions Study Notes

Chapter Outline

• 6.1 Exponential Functions

• 6.2 Graphs of Exponential Functions

• 6.3 Logarithmic Functions

• 6.4 Graphs of Logarithmic Functions

• 6.5 Logarithmic Properties

• 6.6 Exponential and Logarithmic Equations

Introduction to Exponential and Logarithmic Functions

• Focus in on a square centimeter of skin: hundreds of thousands of microscopic organisms (bacteria) present.

• Bacteria reproduce through binary fission, dividing rapidly in optimal conditions (minutes/hours instead of days/years).

Bacterial Growth Example

• Starting with 1 bacterial cell that divides every hour:

  • After 10 hours: 1,024 cells.

  • Extrapolation to 24 hours: over 16 million cells.

6.1 Exponential Functions

• Learning Objectives:

  • Evaluate exponential functions.

  • Find equations of exponential functions.

  • Use compound interest formulas: A = P(1 + \frac{r}{n})^{nt}.

  • Evaluate exponential functions with given base.

Real-World Application Example

• India's Population Growth:

  • In 2021, India has a population of ~1.366 billion, growing by about 1% annually.

  • Projected to exceed China’s population by 2023 due to rapid growth.

Identifying Exponential Functions

• Linear vs Exponential Growth:

  • Linear growth: constant rate of change (e.g., 3 per unit increase).

  • Exponential growth: percentage change per unit time.

Definitions

• Percent change: Change based on a percent of the original amount.

• Exponential growth: Increase based on a constant multiplicative rate of change (e.g., doubling).

• Exponential decay: Decrease based on a constant multiplicative rate.

Example of Exponential Growth vs Linear Growth

• Constructing Functions:

  • Exponential: Starting from 1, doubling outputs (e.g., (y = 2^x), giving outputs 2, 4, 8,…).

  • Linear: Adding a constant e.g., (y = 2 + 2x) (2, 4, 6,…).

  • Comparison shows exponential growth rapidly surpasses linear growth.

General Form

• Exponential Function:

  • Defined as y = ab^x where:

    • a: Initial value.

    • b: Positive base not equal to 1.

    • Growth if b > 1, decay if b < 1.

Characteristics of Exponential Functions

• Domain:

  • All real numbers.

• Range:

  • Positive real numbers for growth; negative for decay.

• y-intercept: 1 when x = 0, horizontal asymptote is 0.

Example: Identifying Exponential Functions

• Given equations, determine if not exponential (base must be a constant):

  • Example: y = x^3 is a power function, not exponential.

Evaluating Exponential Functions

• The base must be positive.

  • Example calculations showing outcome errors when base isn't positive (undefined).

Example Transitioning to Powers

• Evaluate y = 2^x where x = 3 yielding 8.

• Correct order of operations matters.

Exponential Growth Defined

• If the growth rate is proportional to the amount present:

  • Form: y = ae^{kt} where k is positive for growth.

Comparison of Linear vs Exponential Growth Using Model

• Company A: Linear growth: y = 100 + 50t.

• Company B: Exponential growth (e.g., double every year from 100): y = 100(2^t).

• Show graphs differ significantly over time.

Real World Population Example

• India's Population Equation:

  • P(t) = P_0 e^{kt} denoting future predictions.

Fitting Equations Based on Data

• How to Create a Model from two data points.

• Generally forms a unique exponential function.

Compound Interest Calculations

• A = P(1 + \frac{r}{n})^{nt} where:

  • A: Final amount,

  • P: Principal,

  • r: APR as decimal,

  • n: Number of compounding periods.

Applying the Formula for Different Compounding Periods

• A comparison of resulting account values based on compounding frequency.

Example Calculation for Interest and Growth Models

• Illustrate through investments and populations.

Continuous Compounding with e

• A = Pe^{rt} expresses continuous growth/decay well.

Examples of Continuous Models

• Investigate using e and growth charting for better financial forecasting.

Exponential Growth/Decay Basics

• Analyze doubling and half-life behaviors with the simpler functions.

6.2 Graphs of Exponential Functions

• Learning Objectives: Methods to graph, identify domains, ranges, and asymptotes of the exponential curves.

Graph Generating Techniques

• Table creation, plot functions, and basic transformations.

Rapid Growth and Decay Visualization

• Illustrate exponential functions graphically through sigmoidal plots.

Comparison with Linear Models

• Key notes on comparative analysis to understand system behaviors.

Summary of Key Principles

• Critical review of exponential/logarithmic analysis and results.

6.3 Logarithmic Functions

• Learning Objectives: Explore logarithms, relations to exponentials, and conversions between forms.

Conversion Forms

• Use logarithmic functions with practical applications in real-life scenarios.

Defining Logarithmic Interactions

• Important examples on earthquake measures through the Richter scale.

6.4 Graphs of Logarithmic Functions

• Learning Objectives: Exploring logarithmic data modeling.

Transformation Insights

• Deduce the relations between input data and their function outputs for graphing techniques.

6.5 Logarithmic Properties

• Utilize product, quotient, and power rules for logarithms in various mathematical calculations.

Condensing and Synthesizing Logarithmic Expressions

• Key uses of logarithmic properties in problem-solving situations to refine function outputs.

6.6 Exponential and Logarithmic Equations

• Techniques for solving equations involving exponential and logarithmic functions and applications for real-world problems.

Solving Exponential Equations via Logarithms

• General principles discussed for effective mathematical solutions and model fitting using regressions.