Calculus II - Lecture Notes on Natural Log Function (Section 6.1)

Introduction

• Start of Calculus II: Focus on the natural logarithm function.
• Review of properties of natural logarithm and logarithmic functions.

Key Properties of Natural Logarithm

1. Ln(1) = 0

   • Logarithmic property: Any number to the power of 0 is 1.
   • Base of natural logarithms is e (approximately 2.71).

2. Product Property

   • Ln(x * y) = Ln(x) + Ln(y)
   • Can expand or combine logarithms as long as they have the same base.

3. Quotient Property

   • Ln(x/y) = Ln(x) - Ln(y)
   • Same as product property but for division.

4. Power Property

   • Ln(x^r) = r * Ln(x)
   • Exponent can be moved in front of the logarithm.

Practice of Expanding Logarithms

• Example: Expand Ln(3x^4 / (2y^2))
  • Step 1: Apply Quotient Property → Ln(3x^4) - Ln(2y^2)
  • Step 2: Apply Product Property to each term →
    • Ln(3) + Ln(x^4) - (Ln(2) + Ln(y^2))
  • Step 3: Apply Power Property →
    • Result: Ln(3) + 4Ln(x) - Ln(2) - 2Ln(y)

Derivatives and Integrals of Natural Logarithms

• Expanding logarithms simplifies the process of taking derivatives and integrals.
• When expanding logarithms, be cautious of small details, as they can cause significant errors in calculus.

Graph of the Natural Log Function

• Ln(x) is undefined for 0 and negative values.
• The graph approaches the y-axis but never touches it.
• Key point: Ln(1) = 0, crosses the y-axis at (0,1).

Fundamental Theorem of Calculus

• Integral and anti-derivative concepts were briefly discussed.
• Integral of 1/x is Ln |x| + C.

Integration Techniques

• Example: Integral of 1/(5x - 2)
  • Step 1: Substitute U = 5x - 2
  • Step 2: Derive to find du, then replace in the integral.
  • Result: Integral is Ln |5x - 2| + C.

Trigonometric Functions

• Indications on how to handle integrals involving trigonometric functions with logarithms.
• Example: Integral of tan(x) = Ln |sec(x)| + C.

Logarithmic Differentiation

• When faced with complex derivatives, taking the logarithm of both sides can simplify the process.
• Example: Process for differentiating y = (4x^2 + 1)^3 (step through derivative rules).

Conclusion

• Emphasis on practicing properties of logarithms and their derivatives.
• Reminder to learn key trigonometric identities for future integration problems.