Calculus 2 - Lecture on Natural Log Functions (Section 6.1)

Introduction

• Topic: Natural Logarithm (Ln) Functions
• Objective: Review key properties of natural logs, understand their role in derivatives and integrals.

Properties of Natural Logarithms

1. Ln(1) = 0

   • Logarithm base is e (approx. 2.71).
   • Interpretation: e to the power of 0 gives 1.

2. Product Property:

   • Ln(x * y) = Ln(x) + Ln(y).
   • Can expand or combine logarithms using this property.

3. Quotient Property:

   • Ln(x / y) = Ln(x) - Ln(y).
   • Similar to the product property, but for division.

4. Power Property:

   • Ln(x^r) = r * Ln(x).
   • Useful for expanding or combining logarithms.

Logarithms and Calculus

• Expanding Logarithms:
  • Break down complex logarithmic expressions to make derivatives or integrals easier.
• Example: Expand Ln(3x^4 / 2y^2).
  • Solution: Ln(3) + 4Ln(x) - 2Ln(y).

Graphs of Functions

• Graph of e^x:
  • Passes through (0,1) and never crosses the x-axis.
  • As x increases, e^x grows; as x decreases, e^x approaches zero.
• Graph of Ln(x):
  • Inverse of e^x, reflected across y = x.
  • Passes through (1,0), undefined for x ≤ 0.

Integrals and Derivatives

• Integrals (Anti-Derivatives):

  • Reverse operation of derivatives, adding a constant + C to the result.

• Basic Power Rule for Integration:

  • Integral of x^n = (x^(n+1))/(n+1) + C.

• Chain Rule for Derivatives:

  • Derivative of Ln(f(x)) = 1/f(x) * f'(x).

Examples of Calculating Integrals

1. Basic Power Rule:

   • Integral of x^5 = x^6/6 + C.

2. Substitution Method in Integration:

   • Used when direct integration is complex.
   • Example: Integral of sin(x).

Advanced Concepts

• Derivative of 1/x = -1/x^2.
• Special Case Integrals:
  • Integral of 1/x = Ln|x| + C.

Practice Problems

• Expanding Logarithms: Complete practice with provided exercises.
• Combining Logarithms: Use properties to simplify expressions.

Key Takeaways

• Understanding and using the properties of natural logarithms are fundamental in calculus.
• Expand logarithms before integrating or differentiating to simplify the process.
• Remember the graphical behavior of e^x and Ln(x) for better intuition.

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Make sure to practice these concepts and utilize examples provided to reinforce your understanding of natural logarithms in calculus.