Exponential Models and Compound Interest

Exponential change multiplies by the same factor each period, unlike linear change, which adds the same amount.

The factor

A 5% rise means multiplying by 1.05 each period. A 5% fall means multiplying by 0.95. A factor above 1 grows; below 1 decays.

The model

A = P(1 + r)ᵗ, where P is the starting amount, r is the rate as a decimal and t counts periods.

Compounding more often

With n compoundings a year, A = P(1 + r/n)^(nt). More frequent compounding earns slightly more, but the gain shrinks fast.

The limit

Compounding without pause gives A = Pe^(rt), where e ≈ 2.718. It is the ceiling all the others approach.

Solving for time

When the unknown is the exponent, take a logarithm of both sides. That is precisely what logarithms exist to do.

The model and its parts

A = P(1 + r/n)^(nt) gives compound interest: P is the principal, r the annual rate, n the compoundings per year, t the years. Each letter has a meaning, and substituting without knowing which is which produces nonsense.

Continuous compounding

Letting n grow without bound gives A = Pe^(rt). The number e arises as the limit of (1 + 1/n)ⁿ, which is why it turns up in every continuous growth process rather than being an arbitrary constant.

Solving for the exponent needs logs

To find how long money takes to double, the unknown sits in the exponent, so take logs of both sides and use the power rule. This is the standard application and the reason the log laws precede this lesson.

Decay is the same model

Radioactive decay, drug elimination and depreciation use the same formula with a rate below one. Half-life is the time for the quantity to halve, and it is constant regardless of the starting amount — a distinctive signature of exponential decay.