Exponential and Logarithmic Functions Test Review

Introduction

• Condensed review of exponential and logarithmic functions.
• Covers key concepts, common mistakes, and potential test problems.

Exponential Functions

Definition

• Exponential Function: ( f(x) = b^x )
  • b: base
  • x: exponent

Types of Graphs

• Growth Function: ( b > 1 )
  • Graph increases from left to right.
  • Horizontal asymptote (e.g., ( y = 0 )).
• Decay Function: ( 0 < b < 1 )
  • Graph decreases from left to right.
  • Horizontal asymptote.

Key Features

• Y-intercept: ( (0,1) )
• Transformations: Shifts and stretches/compressions can alter position and shape.
• Importance of the base (b) determines growth/decay.

Example Problems

• Identify growth vs. decay by examining the base.
• Example: ( y = \frac{3}{4} ) indicates decay (since ( \frac{3}{4} < 1 )).

Transformations

• General form: ( f(x) = a \cdot b^{(x-c)} + d )
  • a: Vertical stretch/compression
  • c: Horizontal shift
  • d: Vertical shift
• Graphing Techniques: Use transformations on parent graphs (apply shifts, reflect if necessary).

Domain and Range

• Domain: All real numbers ((-\infty, \infty)).
• Range depends on horizontal asymptote.

Logarithmic Functions

Definition and Graph

• Logarithmic Function: Inverse of exponential function.
  • ( y = \log_b{x} )
• General shape: passes through ( (1,0) ).

Transformations

• General form: ( f(x) = a \log_b(x-c) + d )
  • Similar transformations as exponential functions.

Example Transformations

• Identify shifts and reflections.
• Example: ( \ln(x-3) + 1 ) shifts right by 3 and up by 1.

Domain and Range

• Domain: Depends on horizontal shift (e.g., ( (c, \infty) )).
• Range: All real numbers.

Solving Exponential and Logarithmic Equations

Exponential Equations

• One-to-One Property: If bases are equal, set exponents equal.
• Convert to logarithmic form to solve for unknown exponents.

Logarithmic Equations

• Convert to exponential form.
• Check for extraneous solutions.

Evaluating Logarithms

• Convert log to exponential form to evaluate.
• Use change of base formula for non-integer solutions.

Properties of Logarithms

Main Properties

• Product Rule: ( \log_b(mn) = \log_b(m) + \log_b(n) )
• Quotient Rule: ( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) )
• Power Rule: ( \log_b(m^n) = n \log_b(m) )

Applications

• Use properties to condense and expand logarithmic expressions.

Interest Calculations

Simple Interest

• Formula: ( A = P(1 + rt) )

Compound Interest

• Formula: ( A = P\left(1 + \frac{r}{n}\right)^{nt} )
• n: number of compounding periods.

Example Problems

• Compare simple vs. compound interest for investments.
• Recognize impact of compounding frequency and interest rate.

Conclusion

• Reviewed key concepts and transformations related to exponential and logarithmic functions.
• Emphasized importance of understanding both function behaviors and graphing strategies.
• Understanding relationships and transformations is crucial for solving equations and applying properties effectively.