Overview
The lecture demonstrates how to solve equations involving natural logarithms, specifically finding the solutions for ln(x)² = 1.
Solving ln(x)² = 1
- ln(x) means the natural logarithm, or logarithm with base e.
- The equation given is (ln(x))² = 1.
- Take the square root of both sides, yielding ln(x) = ±1.
- This creates two separate equations: ln(x) = 1 and ln(x) = –1.
Solving for x
- When ln(x) = 1, rewrite as e^(ln(x)) = e^1, so x = e.
- When ln(x) = –1, rewrite as e^(ln(x)) = e^(–1), so x = 1/e.
- The solutions are x = e or x = 1/e.
Alternative Solution Method
- You can raise both sides of the equation to the exponent base e, getting x = e^(±1).
- This simplifies to the same solutions: x = e or x = 1/e.
Notation for Solution Sets
- Solution sets can be listed with curly brackets: x ∈ {e, 1/e}.
- This notation means x is an element of the set containing e and 1/e.
Key Terms & Definitions
- Natural logarithm (ln) — logarithm with base e (where e ≈ 2.718).
- Exponentiation — raising a number to a specific power.
- Solution set notation (curly brackets) — lists all possible solutions, e.g. {a, b} means x = a or x = b.
Action Items / Next Steps
- Practice solving similar logarithmic equations.
- Review notation for solution sets and intervals.