Exponential models

Exponential growth and decay are modelled by P = Aeᵏᵗ. A is the starting value, when t = 0. If k > 0 the quantity grows; if k < 0 it decays.

To find k from data, substitute a known value and use ln. If P = 850eᵏᵗ and P = 1200 when t = 3, then 1200/850 = e³ᵏ, so k = ln(1200/850) ÷ 3 = 0.115. Equations like 3²ˣ + 5 × 3ˣ − 6 = 0 are quadratics in 3ˣ: (3ˣ + 6)(3ˣ − 1) = 0 gives 3ˣ = 1, so x = 0.