Chapter 4. Exponents, Logarithms and Exponential Growth

Laws of exponents (a, b ≠ 0) [3][4]:

aᵐ · aⁿ = aᵐ⁺ⁿ

aᵐ / aⁿ = aᵐ⁻ⁿ

(aᵐ)ⁿ = aᵐⁿ

a⁰ = 1 and a⁻ⁿ = 1/aⁿ

a^(1/n) = ⁿ√a

Logarithms. log_b(x) = y means bʸ = x. Key properties:

log_b(xy) = log_b x + log_b y

log_b(x/y) = log_b x − log_b y

log_b(xⁿ) = n log_b x

Change of base: log_b x = ln x / ln b

The natural logarithm ln uses the base e ≈ 2.71828.

Exponential models. Growth or decay: y = y₀ · bᵗ. Continuous growth: y = y₀ e^(kt). Compound interest: A = P(1 + r/n)^(nt), where P is principal, r the annual rate, n the number of compounding periods per year and t the number of years.

Review: (a) Evaluate log₂ 32. (Answer: 5.) (b) Solve 2ˣ = 20 for x. (Answer: x = log 20 / log 2 ≈ 4.32.)