Lecture Notes: Exponential and Logarithmic Functions

Introduction to Chapter 4

• Focus on exponential and logarithmic functions.
• Applications in science and engineering.

Exponential Functions

Definition

• Form: ( a^x )
  • Base ( a ) must be greater than 0.
  • ( x ) typically a rational number but can be irrational.

Characteristics

• Base ( a ) dictates growth or decay.
  • ( a > 1 ): exponential growth.
  • ( 0 < a < 1 ): exponential decay.
  • Positive or negative exponent affects growth or decay.

Examples

• ( 2^x ) grows much faster than quadratic ( x^2 ).
• Irrational exponents (e.g., ( 5^{\sqrt{3}} )) result in irrational numbers.
• Important to maintain precision to avoid round-off errors.

Graphing

• ( a^x ): exponential growth if ( a > 1 ), decay if ( 0 < a < 1 ).
• Domain: ( (-\infty, +\infty) )
• Range: ( (0, +\infty) )

Exponential vs Polynomial Growth

• Exponentials (( a^x )) grow faster than polynomials (e.g., ( x^2 )).
• Cross points differ (e.g., ( 2^x ) vs ( x^2 )).

Applications

Compound Interest

• Formula: [ A = P \left(1 + \frac{r}{n}\right)^{nt} ]
  • ( P ): principal
  • ( r ): annual interest rate
  • ( n ): number of compounding periods per year
  • ( t ): time in years
• Compounding increases future value.
  • More frequent compounding (monthly, daily) yields more return.
• Continuous compounding reaches a limit.

Annual Percentage Rate (APR)

• Adjusts the simple interest rate for the number of compounding periods.
• APR is less than the nominal rate due to more frequent compounding.

Conclusion

• Understanding of exponential functions is crucial for grasping logarithms.
• Exponential functions have unique growth patterns important in financial calculations like compound interest.

Important Tips

• When calculating with exponentials, carry as many decimals as possible to reduce errors.
• Recognize the influence of base and sign of the exponent on the function's behavior.