Learn & Review: Calculus 2 Full College Course

Calculus II: Section 6.1 - The Natural Logarithm Function

This section reviews and expands upon the properties of the natural logarithm function, demonstrating how these properties are crucial for simplifying derivatives and integrals. It also covers the graphs of exponential and natural logarithm functions and introduces basic integration techniques involving the natural logarithm.

I. Properties of Natural Logarithms

The natural logarithm, denoted as ln(x), is a logarithm with base e (Euler's number, approximately 2.718). The following properties are fundamental:

• Property 1: ln(1) = 0

  • This stems from the definition of logarithms: base^power = argument. For ln(1), e^0 = 1.

• Property 2: Product Property - ln(xy) = ln(x) + ln(y)

  • A product within a logarithm can be expanded into the sum of two logarithms.
  • This property is bidirectional; two logarithms with the same base can be combined into a single logarithm of their product.

• Property 3: Quotient Property - ln(x/y) = ln(x) - ln(y)

  • Division within a logarithm can be expanded into the difference of two logarithms.
  • Similar to the product property, this allows combining logarithms by subtraction.

• Property 4: Power Property - ln(x^r) = r * ln(x)

  • An exponent within the argument of a logarithm can be moved to the front as a coefficient.
  • This property is key for expanding or combining logarithmic expressions.

Expanding Logarithms: Examples

The lecture emphasizes working from the "outside in" when expanding complex logarithmic expressions, applying properties step-by-step.

• Example 1: ln(3x^4 / (2y^2))

  1. Quotient Rule: ln(3x^4) - ln(2y^2)
  2. Product Rule: (ln(3) + ln(x^4)) - (ln(2) + ln(y^2))
  3. Power Rule: ln(3) + 4*ln(x) - (ln(2) + 2*ln(y))
  4. Distribute Negative: ln(3) + 4*ln(x) - ln(2) - 2*ln(y)
  • Caution: Be mindful of parentheses and distributing negative signs, especially when expanding quotients or differences. Exponents can only be moved forward if they apply to the entire argument of the logarithm.

• Example 2: ln(x^2 + 1) / sqrt(x)

  1. Rewrite Root: ln((x^2 + 1) / x^(1/2))
  2. Quotient Rule: ln(x^2 + 1) - ln(x^(1/2))
  3. Power Rule: ln(x^2 + 1) - (1/2)*ln(x)
  • Note: The term ln(x^2 + 1) cannot be further expanded because addition within the logarithm is not a property.

Combining Logarithms: Example

The process of combining logarithms is the reverse of expanding. Coefficients are moved to become exponents, and sums/differences are converted back to products/quotients.

• Example: Combine ln(x^5) + ln(x^2 + 1)^(1/2) - ln(x - 2)^4
  1. Move Coefficients to Exponents: ln(x^5) + ln(sqrt(x^2 + 1)) - ln((x - 2)^4)
  2. Combine using Product/Quotient Rules: ln( (x^5 * sqrt(x^2 + 1)) / (x - 2)^4 )
  3. Apply Overall Coefficient (if any): If there was a coefficient like 2/3 in front, it would become the exponent of the entire combined logarithm.

II. Graphs of Exponential and Natural Logarithm Functions

• y = e^x:
  • Always positive output.
  • Crosses the y-axis at (0, 1).
  • Increases rapidly for positive x, approaches 0 for large negative x (asymptotic to the x-axis).
• y = ln(x):
  • The inverse function of y = e^x.
  • Graphically, it's a reflection of y = e^x across the line y = x.
  • Crosses the x-axis at (1, 0).
  • Undefined for x ≤ 0 (asymptotic to the y-axis).
  • Increases slowly for large positive x.

III. Basic Integration Rules

• Power Rule for Integration: ∫x^n dx = (x^(n+1))/(n+1) + C (for n ≠ -1)
• Constant Multiple Rule: ∫c*f(x) dx = c*∫f(x) dx
• Integral of Sine/Cosine:
  • ∫sin(x) dx = -cos(x) + C
  • ∫cos(x) dx = sin(x) + C
• Integral of 1/x: This is a special case where the power rule doesn't apply (n = -1).
  • ∫(1/x) dx = ln|x| + C
  • The absolute value is crucial because the domain of ln(x) is x > 0, but the function 1/x is defined for negative x as well.

Integration Techniques Introduced:

• Separating Terms: A single term in the denominator can be used to split a fraction into multiple terms, simplifying integration.
  • Example: ∫(x^4 - 1) / x^2 dx becomes ∫(x^2 - x^-2) dx.
• Substitution (u-substitution): Used when the integrand is not directly in the integration table.
  1. Identify a suitable substitution u (often an inner function).
  2. Find du/dx and solve for dx.
  3. Substitute u and dx into the integral, ensuring all x variables are eliminated.
  4. Integrate with respect to u.
  5. Substitute back to express the result in terms of the original variable (x).
• Trigonometric Integrals:
  • Identities: Crucial for rewriting trigonometric expressions to fit known integration rules.
  • Tangent: ∫tan(x) dx = ln|sec(x)| + C (derived using u-substitution with u = cos(x)). This can also be written as -ln|cos(x)| + C.
  • Secant: ∫sec(x) dx = ln|sec(x) + tan(x)| + C (derived using a special multiplication trick).
  • Other Trig Integrals: Formulas for cot(x), csc(x), and their variations are provided, often derived similarly to tan(x) and sec(x).

IV. Derivatives Involving Natural Logarithms

• Basic Derivative: d/dx [ln(x)] = 1/x
• Chain Rule Application: When the argument of the natural logarithm is a function g(x) instead of just x, the chain rule applies.
  • d/dx [ln(g(x))] = (1 / g(x)) * g'(x) or g'(x) / g(x)
  • This means: "one over the inside, times the derivative of the inside."

Examples of Logarithmic Derivatives:

• d/dx [ln(3x^2 - 1)] = (6x) / (3x^2 - 1)
• d/dx [ (ln(x))^(1/3) ] involves the general power rule first, then the derivative of ln(x): (1/3) * (ln(x))^(-2/3) * (1/x)
• d/dx [ln(x^2 + 1)^(1/2)] requires expanding first (if possible) or applying the chain rule carefully.
• d/dx [ln( (x^2+1) / sqrt(x) )] is easier after expanding using log properties: d/dx [ln(x^2+1) - (1/2)ln(x)].

V. Logarithmic Differentiation

This technique is used to find derivatives of complex functions, especially those involving products, quotients, and powers that would otherwise require lengthy application of the product, quotient, and chain rules.

1. Take the Natural Logarithm: Apply ln() to both sides of the equation y = f(x).
2. Expand: Use logarithm properties to expand the ln(y) side as much as possible.
3. Differentiate Implicitly: Take the derivative of both sides with respect to x. Remember that d/dx [ln(y)] = (1/y) * dy/dx.
4. Solve for dy/dx: Isolate the derivative term (dy/dx or y').
5. Substitute Back: Replace y with its original expression in terms of x.

• Example: Finding the derivative of y = (x-1)^4 / (2x-1)^(1/3)
  1. ln(y) = ln((x-1)^4) - ln((2x-1)^(1/3))
  2. ln(y) = 4*ln(x-1) - (1/3)*ln(2x-1)
  3. d/dx[ln(y)] = d/dx[4*ln(x-1) - (1/3)*ln(2x-1)]
  4. (1/y) * dy/dx = 4/(x-1) - (1/3)*(2/(2x-1))
  5. dy/dx = y * [ 4/(x-1) - 2/(3(2x-1)) ]
  6. Substitute y back: dy/dx = [(x-1)^4 / (2x-1)^(1/3)] * [ 4/(x-1) - 2/(3(2x-1)) ]

VI. Integrals Involving Natural Logarithms

• Core Rule: ∫(1/u) du = ln|u| + C
• Application: This rule is fundamental. Often, substitution is required to transform an integral into the form ∫(1/u) du.
  • Example: ∫(1 / (5x - 2)) dx
    1. Let u = 5x - 2, so du = 5 dx, which means dx = du/5.
    2. The integral becomes ∫(1/u) * (du/5) = (1/5) ∫(1/u) du.
    3. Integrating gives (1/5) ln|u| + C.
    4. Substituting back: (1/5) ln|5x - 2| + C.
  • Example with Trig: ∫(sin(x) / cos(x)) dx (i.e., ∫tan(x) dx)
    1. Let u = cos(x), so du = -sin(x) dx, which means dx = du / (-sin(x)).
    2. The integral becomes ∫(sin(x) / u) * (du / (-sin(x))) = -∫(1/u) du.
    3. Integrating gives -ln|u| + C.
    4. Substituting back: -ln|cos(x)| + C, which is equivalent to ln|sec(x)| + C.

VII. New Trigonometric Integrals

Based on the derivation of ∫tan(x) dx and ∫sec(x) dx, the following integration rules are added to the table:

• ∫tan(u) du = ln|sec(u)| + C
• ∫sec(u) du = ln|sec(u) + tan(u)| + C
• ∫cot(u) du = ln|sin(u)| + C
• ∫csc(u) du = ln|csc(u) - cot(u)| + C

These rules are essential for integrating various trigonometric expressions, often requiring substitution and trigonometric identities.