Logarithm Properties and Applications

Overview

This lecture introduces natural logarithms (ln), reviews their properties, and demonstrates their use in derivatives and integrals, including practice with algebraic manipulation, chain rule, and techniques for integration involving substitution and trigonometric identities.

Natural Logarithm Properties

• ln(1) = 0 because e⁰ = 1.
• Product Property: ln(xy) = ln(x) + ln(y).
• Quotient Property: ln(x/y) = ln(x) - ln(y).
• Power Property: ln(x^r) = r·ln(x).
• Logarithm properties can be used to expand or combine expressions, but only products, quotients, and powers—not sums.

Expanding and Combining Logarithms

• Expand logarithmic expressions by separating products, quotients, and moving exponents forward.
• Combine logarithms by reversing the above properties, ensuring all logs have the same base.
• Only use listed logarithm properties; do not invent new ones.

Fundamental Graphs and Domains

• Graph of e^x is always positive; never crosses the x-axis.
• ln(x) is only defined for x > 0; ln(0) and ln(negative) are undefined.
• ln(x) and e^x are inverse functions, reflecting over y = x.

Integration and Differentiation Rules

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1.
• ∫1/x dx = ln|x| + C.
• Derivative of ln(x) is 1/x; for ln(f(x)), use chain rule: d/dx[ln(f(x))] = f'(x)/f(x).
• Absolute value is required in ln|x| after integration.

Chain Rule and Product Rule with ln(x)

• For y = ln(g(x)), derivative = g'(x)/g(x).
• For products like x³·ln(5x), use product rule: derivative of first·second + first·derivative of second.
• For ln of a composite function, always apply the chain rule.

Logarithmic Differentiation & Implicit Differentiation

• Take ln of both sides to simplify differentiation of complicated products or quotients.
• After expanding using log properties, differentiate both sides, applying the chain rule to ln(y).
• Isolate dy/dx in implicit problems, substitute y back in terms of x for explicit derivative.

Integration by Substitution and Trigonometric Integrals

• For integrals not matching a table, use substitution to convert to a basic form (e.g., ∫1/(5x-2) dx with u = 5x-2).
• Substitution often uses derivatives present in the integrand.
• Integrals of trigonometric functions may require identities and log relationships (e.g., ∫tan(x) dx = -ln|cos(x)| + C).
• For secant and cosecant integrals, use special tricks and substitutions to relate them to natural logs.

Key Terms & Definitions

• Natural Logarithm (ln x) — Logarithm base e (~2.718), defined only for x > 0.
• Product/Quotient/Power Properties — Rules for expanding/combining logs involving multiplication, division, and exponents.
• Chain Rule — Differentiation rule: derivative of f(g(x)) is f'(g(x))·g'(x).
• Logarithmic Differentiation — Technique of taking ln on both sides to simplify differentiation.
• Integration by Substitution — Rewriting variables to match integration table forms.

Action Items / Next Steps

• Practice expanding and combining logarithmic expressions.
• Complete derivative and integral problems involving ln(x) and composite functions.
• Review trigonometric identities for upcoming integration techniques.
• Read Section 6.2 before next class.