Overview
This lecture covers the basics of exponentiation, exponential functions, inverse functions, and logarithmic functions, including their properties, graphs, and relationships.
Exponentiation Review
- bⁿ means multiplying n copies of b when n is a positive integer.
- b^(p/q) = q-th root of (b^p) for rational exponents and b > 0.
- For f(x) = bˣ (b > 0), exponent rules must hold for all real x, y:
- bˣ × bʸ = b^(x+y)
- bˣ / bʸ = b^(x−y)
- (bˣ)ʸ = b^(xy)
- bˣ is always positive.
Exponential Functions
- f(x) = bˣ (b > 0) has domain: all real x, range: y > 0.
- f(0) = b⁰ = 1 for any b.
- If b > 1, f(x) increases; if b < 1, f(x) decreases.
- The natural exponential function is f(x) = eˣ, where e ≈ 2.71828 and is always increasing.
- The graph of eˣ has a y-intercept at (0, 1).
Inverse Functions
- An inverse function f⁻¹ reverses f: if y = f(x), then f⁻¹(y) = x.
- f⁻¹(f(x)) = x and f(f⁻¹(y)) = y, domains considered.
- Function f has an inverse if it is one-to-one (every output has one input).
- Horizontal line test: every horizontal line crosses f(x) at most once if f is invertible.
- To find f⁻¹: Solve y = f(x) for x, then switch x and y.
- Example: If f(x) = x⁴ + 1 (x ≥ 0), then f⁻¹(x) = fourth root of (x − 1), domain [1, ∞).
Graphs of Inverse Functions
- The graph of f⁻¹(x) is the reflection of f(x) across the line y = x.
Logarithmic Functions
- If b > 0 and b ≠ 1, f(x) = bˣ is one-to-one, domain: (−∞, ∞), range: (0, ∞).
- The inverse is log base b: y = log_b(x) ⇔ bʸ = x, domain: (0, ∞), range: (−∞, ∞).
- Graphs of bˣ and log_b(x) are reflections across y = x.
Properties of Logarithms
- log_b(xy) = log_b(x) + log_b(y)
- log_b(x/y) = log_b(x) − log_b(y)
- log_b(xᶻ) = z × log_b(x)
- log_b(1) = 0, log_b(b) = 1
- For x > 0, b^(log_b(x)) = x; for any x, log_b(bˣ) = x
Natural Logarithm (ln)
- Natural log: ln(x) is the inverse of eˣ; ln(1) = 0, ln(e) = 1.
- ln(x) domain: (0, ∞), range: (−∞, ∞).
- ln(eˣ) = x for any real x; e^(ln(x)) = x for x > 0.
- Graphs of eˣ and ln(x) are reflections across y = x.
Example: Solving with Natural Logarithm
- To solve 500e^(−0.4t) = 10:
- Divide by 500: e^(−0.4t) = 1/50.
- Take ln: ln(1/50) = −0.4t.
- t = ln(1/50)/(−0.4) ≈ 9.78 seconds.
Change of Base Formula
- log_b(x) = log_c(x) / log_c(b) for any b, c > 0, b ≠ 1, c ≠ 1.
- Using natural log: log_b(x) = ln(x) / ln(b).
Key Terms & Definitions
- Exponentiation — Repeated multiplication of a base.
- Exponential Function — Function of the form f(x) = bˣ, b > 0.
- Inverse Function — Function that reverses another: f⁻¹(f(x)) = x.
- One-to-One Function — Every output is from a single input.
- Logarithm (log_b(x)) — Inverse of the exponential function; yields exponent.
- Natural Exponential (eˣ) — Exponential with base e ≈ 2.71828.
- Natural Logarithm (ln(x)) — Logarithm with base e.
Action Items / Next Steps
- Practice finding and verifying inverse functions.
- Solve additional logarithmic and exponential equations.
- Review properties of logarithms and exponent rules.