Continuous growth and the base e
The natural exponential eˣ is special because its derivative is itself. In a continuous growth model, dy/dt = ky has solution y(t) = y₀eᵏᵗ. Positive k gives growth; negative k gives decay. The doubling time for k > 0 is ln(2)/k, and the half-life for k < 0 is ln(2)/|k|.
Exponential growth eventually outpaces any fixed power xⁿ, but the phrase “eventually” matters: over a limited range, a polynomial can be larger. Real systems also have constraints, so indefinite exponential growth is rarely a realistic long-term model. See the normal distribution for a contrasting bell-shaped model and logarithms for linearising multiplicative change.
Turn a repeated percentage into a model
If a quantity grows by 5% each month, each month multiplies it by 1.05. Starting at 100, after t months it is 100(1.05)ᵗ; after two months it is 110.25, not 110, because the second increase applies to the updated value. A decay of 5% per step multiplies by 0.95. This constant-factor-per-step property distinguishes exponential change from a constant additive increase, which is linear.
In continuous time, the differential equation dy/dt=ky says the instantaneous rate is proportional to the current amount. Separating variables and solving gives y=y₀eᵏᵗ. Here k is a continuous rate per unit time; the doubling time is ln(2)/k when k>0. A discrete growth factor b and continuous rate k are related by b=eᵏ for one time step. Always state the time unit and whether the model compounds in steps or continuously. Exponential models can describe early population growth or radioactive decay, but resources, changing rates, and background levels can make long-term extrapolation unrealistic.
Try this above
Change the base from below 1 to above 1. Compare equal horizontal steps and observe the constant multiplication factor; then compare the curve with its inverse.
The inverse relationship with logarithms lets you solve for time or rate. If 100 grows to 200 with a 5% monthly factor, solve 100(1.05)^t=200, giving t=ln 2/ln 1.05, about 14.2 months. This is a model-based estimate and assumes the rate remains constant. In continuous growth y=y₀eᵏᵗ, k is not the same as a 5% discrete rate; the equivalent continuous rate is ln(1.05) per month. State which convention is being used when comparing rates.
A discrete model yₙ=y₀bⁿ has constant ratio yₙ₊₁/yₙ=b, while a linear model has constant difference yₙ₊₁−yₙ. This gives a practical way to distinguish multiplicative from additive change in equally spaced observations. Real measurements include noise, so compare ratios on a suitable scale and check residuals rather than infer exponential growth from a curve that merely rises quickly. Logs can turn a positive exponential relation into a linear relation for analysis.