Any quantity that changes by a fixed percentage every fixed time interval — rather than by a fixed amount — follows an exponential model. A population growing 3% per year, an investment earning 5% interest, a radioactive sample losing half its mass every few hours, a car losing 15% of its value every year: all of these use the same underlying structure. The skill in this lesson is translation. You read a sentence such as "the colony triples every 4 hours" or "the drug level falls by 18% each hour", turn it into an equation, use the equation to predict, and, when the unknown is the time, solve an exponential equation to find when something happens.
The general growth/decay model
Where does come from? If a population of grows by 3%, the new population is the old one plus 3% of it: . Each year multiplies by the same , so after years you have multiplied by a total of times: . For decay expressed as a percentage decrease, subtract instead: with (e.g. 15% depreciation means the base is , because 85% of the value survives each year). The base is the fraction that remains, never the rate by itself.
| The words say | Model | Base per period |
|---|---|---|
| grows 3% per year | 1.03 | |
| depreciates 15% per year | 0.85 | |
| doubles every 6 hours | 2 per 6 hours | |
| triples every 4 years | 3 per 4 years | |
| half-life of 5 hours | one half per 5 hours |
Half-life and doubling time
Some situations are described directly in terms of how long it takes to halve or double, rather than as a percentage rate. These use the same structure, but with the base built directly from the description. The exponent counts how many half-lives have passed: after 15 hours with a 5-hour half-life, half-lives have passed, so the amount has been halved three times, . The same logic works for doubling, tripling, or any "multiplies by every time units" pattern: . Carbon-14 dating, medication dosing and bacterial cultures in a lab all use exactly this form.
Solving exponential equations by matching bases
If two exponential expressions with the same base are equal, then their exponents must be equal too — this lets you solve exactly, without logarithms, whenever both sides can be rewritten using a common base. It works because is always increasing or always decreasing, so it never gives the same output twice. This is how you answer "when?" questions. For an 80 mg sample with a 5-hour half-life, when is 10 mg left? Set , divide by 80 to isolate the power, , match exponents, , so hours.
- Isolate the power: undo any addition or multiplication outside it first. In , subtract 4 and divide by 3 to get .
- Write both sides with a common base. Know your powers: ; ; . Fractions and roots count too: and .
- Simplify with the power-of-a-power law, e.g. .
- Set the exponents equal and solve the resulting linear equation.
- Check by substituting back, and make sure the answer makes sense in context (a negative time or a fraction of a person needs a second look).
When there is no common base
Some equations cannot be matched: ("when will a 3% population double?") has no nice common base. In this course, solve these by systematic trial or from a graph or table: and , so it takes a little over 23 years. (Logarithms, which solve these exactly, arrive in Grade 12.) Whatever the method, keep in the same unit as the rate: a monthly rate needs in months, and a half-life in hours needs in hours. Finally, ask whether the answer is reasonable. A 3% population should take a couple of decades to double (the "rule of 72" estimate is years), a quantity with a 5-hour half-life should be under half its size after 6 hours, and a depreciating car can approach but never reach zero dollars. A model is also only trustworthy over a sensible range: no real population grows at 3% forever, because food and space run out.