Exponentials, Logarithms & Growth Models
Exponential functions model compound interest, population growth, radioactive decay, and viral spread. Logarithms invert them and appear in information theory, acoustics, and algorithm complexity.
Key Concepts
Exponential Growth & Decay
P(t) = P₀ · aᵗ. If a > 1: growth. If 0 < a < 1: decay. Natural base: e ≈ 2.71828. P(t) = P₀eᵏᵗ. Doubling time: t = ln(2)/k. Half-life: t = ln(0.5)/k.
Laws of Logarithms
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).
Compound Interest
A = P(1 + r/n)^(nt) for n compoundings per year. Continuous: A = Peʳᵗ. At r=5% for 10 years: compound monthly vs. continuous differs by very little.
Solving Exponential Equations
2^x = 32 → x = log₂(32) = 5. General: aˣ = b → x = log_a(b) = ln(b)/ln(a). Take log of both sides to bring exponent down.
Live Python Practice
Interactive Lab
Check Your Understanding
log₂(64) equals:
The natural number e ≈
For compound interest A = Peʳᵗ, what does t represent?