Exponential Functions, Equations, and Inequalities | General Mathematics

Introduction

This tutorial will guide you through the fundamentals of exponential functions, equations, and inequalities. Understanding these concepts is crucial in mathematics, as they are widely applicable in fields such as finance, biology, and physics. By the end of this guide, you will have a solid grasp of how to work with exponential expressions and solve related problems.

Step 1: Understanding Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent.

• Definition: An exponential function can be expressed as ( f(x) = a \cdot b^x ), where:
  • ( a ) is a constant (the initial value).
  • ( b ) is the base (a positive constant).
  • ( x ) is the exponent (a variable).
• Graphing: The graph of an exponential function is characterized by:
  • Rapid growth (if ( b > 1 )) or decay (if ( 0 < b < 1 )).
  • A horizontal asymptote at ( y = 0 ).

Practical Tip

Identify the base and the initial value when graphing or solving exponential functions to understand their behavior.

Step 2: Working with Exponential Equations

Exponential equations involve expressions where the variable is in the exponent.

• Form: Typically structured as ( a \cdot b^x = c ).
• Solving Steps:
  1. Isolate the exponential expression: ( b^x = \frac{c}{a} ).
  2. Use logarithms to solve for ( x ):
     • Apply the logarithmic identity: ( x = \log_b\left(\frac{c}{a}\right) ).
• Example: Solve ( 2^x = 16 ).
  • Isolate: ( 2^x = 2^4 ).
  • Therefore, ( x = 4 ).

Common Pitfall

Ensure the bases are the same when equating exponential terms. If not, use logarithms to solve.

Step 3: Exploring Exponential Inequalities

Exponential inequalities involve expressions where the variable is in the exponent and can be represented as ( a \cdot b^x < c ) or ( a \cdot b^x > c ).

• Steps to Solve:
  1. Isolate the exponential expression.
  2. Use logarithms to transform the inequality:
     • For ( b^x < d ), take ( \log_b ) of both sides.
     • For example, if ( 2^x < 16 ):
       • Isolate: ( 2^x < 2^4 ).
       • Thus, ( x < 4 ).
  3. Pay attention to the base when solving:
     • If ( 0 < b < 1 ), the direction of the inequality will flip.

Practical Tip

Graph the corresponding exponential function to visualize the solution set for inequalities.

Step 4: Real-World Applications

Exponential functions and inequalities are used in various scenarios, including:

• Finance: Calculating compound interest.
• Population Growth: Modeling how populations increase over time.
• Radioactive Decay: Understanding how substances decrease over time.

Example Application

For compound interest, the formula is given by: [ A = P(1 + \frac{r}{n})^{nt} ] where:

• ( A ) is the amount of money accumulated after n years, including interest.
• ( P ) is the principal amount (initial investment).
• ( r ) is the annual interest rate (decimal).
• ( n ) is the number of times that interest is compounded per year.
• ( t ) is the time in years.

Conclusion

In this tutorial, you learned about exponential functions, how to solve exponential equations, and how to tackle exponential inequalities. Remember to practice these concepts with real-world examples to enhance your understanding. As you progress, consider exploring more advanced topics such as logarithmic functions and their applications in different fields. Happy learning!