Page 2: Logarithmic Laws and Properties

The second page delves into the essential laws and properties of logarithms, providing a foundation for solving logarithmic equations.

Definition: The logarithm of a number is the power to which a base must be raised to produce that number.

Highlight: Key logarithmic laws include:

• Multiplication law: logₐ(x×y) = logₐx + logₐy
• Division law: logₐ(x÷y) = logₐx - logₐy
• Power law: logₐ(xⁿ) = n×logₐx

Example: If 3² = 9, then log₃9 = 2, demonstrating the relationship between exponentials and logarithms.

Page 3: Mixed Exercises and Applications

The final page presents various exercises applying logarithmic and exponential concepts, demonstrating practical problem-solving techniques.

Example: Solving 4ˣ = 2³ involves using logarithms:

1. Take log of both sides
2. Apply power law
3. Solve for x using log properties

Highlight: When solving logarithmic equations, always check solutions as logarithms of negative numbers are undefined.

Vocabulary: The term "linear data" refers to data that forms a straight line when plotted, often achieved by applying logarithmic transformations to exponential relationships.

Page 1: Exponential Functions and Natural Logarithms

This page introduces fundamental concepts of exponential functions and their relationship with logarithms. The content explores the basic properties and behaviors of these mathematical functions.

Definition: An exponential function f$x$ = aˣ is a function where a constant is raised to a variable power.

Highlight: All exponential functions pass through the point (0,1) on the coordinate plane and approach zero as x decreases.

Example: For the function f$x$ = eˣ, its derivative f'$x$ = eˣ, demonstrating the unique property of the natural exponential function.

Vocabulary: The number 'e' is a special mathematical constant approximately equal to 2.71828, often called Euler's number.