Exponential decay describes any quantity that loses a fixed percentage of itself each period. It uses the same recursive multiplication structure as exponential growth, but the multiplier is below 1. Caffeine in the bloodstream, radioactive isotopes, capacitor voltage and depreciating assets all decay exponentially.
The exponential decay formula
Continuous form: x(t) = x₀ × e-kt. Discrete form: x(t) = x₀ × (1 - r)t. The two are linked by k = -ln(1 - r), so a 10% per-period decay rate corresponds to k ≈ 0.1054.
Half-life
Half-life t½ = ln(2) / k is the time for a quantity to fall to 50% of its starting value. Carbon-14 has a half-life of 5,730 years; caffeine about 5 hours; ibuprofen about 2 hours; uranium-238 about 4.47 billion years. To solve specifically for t½ and the remaining amount at any elapsed time, see the half-life calculator.
The decay constant k
The decay constant is the instantaneous fractional loss per unit time. Larger k means faster decay. The relationship k = ln(2) / t½ lets you convert between the two whenever you know one.
Radioactive decay
Radiometric dating uses the known decay constants of isotopes (carbon-14, potassium-40, uranium-238) to back-calculate the age of a sample. Measure the remaining fraction, take the natural logarithm and divide by -k to get elapsed time. For example, a bone sample containing 68% of the carbon-14 found in living tissue implies an elapsed time of t = -ln(0.68) / k, where k = ln(2) / 5,730 ≈ 0.000121 per year, giving t ≈ 3,190 years.
Drug metabolism
Pharmacokinetics models blood concentration with the same equation. After 5 half-lives, only about 3.1% of the original dose remains, so steady-state drug levels stabilize after roughly 5 half-life intervals on a repeating dose. If you only have two measured blood concentrations rather than a known rate, use the decay rate calculator to back out k first, then plug it in here.
Worked example
A 200 mg dose of caffeine with a 5-hour half-life: k = ln(2) / 5 = 0.1386 per hour. After 10 hours, remaining = 200 × e^(-0.1386 × 10) = 50 mg, exactly two half-lives as expected.
More everyday decay examples
Newton's law of cooling follows the same equation, with the temperature difference from ambient playing the role of x. A cup of coffee at 90°C in a 20°C room starts with a 70°C difference; with a cooling constant of k = 0.05 per minute, after 20 minutes that difference falls to 70 × e^(-1), about 25.75°C, putting the coffee at roughly 45.75°C. A capacitor discharging through a resistor follows the same curve with RC in place of 1/k: a 10 microfarad capacitor through a 100k ohm resistor has RC = 1 second, so after 3 seconds the voltage falls to e^(-3), about 4.98% of its starting value.
Declining-balance depreciation, which removes a fixed percentage of the remaining book value each year, is exponential decay in disguise. A $30,000 vehicle depreciating at 15% per year under this method is worth 30,000 × (0.85)^5, about $13,311, after 5 years, noticeably more than straight-line depreciation, which subtracts a flat dollar amount every year, would leave over the same period.
A quick sanity check
A simple way to check any decay result by eye is the halving table: after 1 half-life, 50% remains; after 2, 25%; after 3, 12.5%; after 4, 6.25%; after 5, about 3.1%; and after 10 half-lives, under 0.1% remains. Comparing a calculated remaining percentage against this table catches most sign errors or misplaced decimal points before they lead to a wrong conclusion.
FAQ
What is exponential decay?
Exponential decay describes a quantity that shrinks by the same fixed percentage every period, modeled as x(t) = x₀ × e^(-kt). Because the rate of loss is proportional to whatever amount remains, decay is fastest at the start and gradually slows as the quantity falls, so the curve never quite reaches zero. Radioactive isotopes and drug elimination both follow this exact pattern.
What is the exponential decay formula?
There are two equivalent forms. The continuous form is x(t) = x₀ × e^(-kt), using decay constant k. The discrete form is x(t) = x₀ × (1 - r)^t, using per-period decay rate r. Here x₀ is the starting amount and t is elapsed time. For a 10% per-period rate, k works out to about 0.1054, since the two constants are linked.
What is the half-life formula?
Half-life, written t½, equals ln(2) divided by the decay constant k, so t½ = ln(2) / k. It is the fixed amount of time needed for any quantity to fall to exactly half its starting value, regardless of how much you started with. Carbon-14, with k around 1.21 × 10⁻⁴ per year, has a half-life of about 5,730 years.
How do I calculate exponential decay?
Multiply the starting amount by e raised to the power of negative k times t, or by (1 - r) raised to the power t if you know the per-period rate instead. For example, 100 mg decaying at k = 0.1 per hour leaves 100 × e^(-0.5), about 60.65 mg, after 5 hours, roughly 61% of the original dose remaining.
What is the difference between half-life and decay constant?
The decay constant k measures the instantaneous fractional loss per unit time, while half-life is the time needed for the amount to drop to 50%. A larger k always means a shorter half-life. The two describe the same physical process from different angles and convert directly with t½ = ln(2) / k.
Where does exponential decay appear in real life?
Exponential decay shows up in radioactive isotopes losing atoms, drugs clearing from the bloodstream, capacitors discharging their voltage, atmospheric pressure thinning with altitude, hot objects cooling under Newton's law, and depreciating assets losing resale value. In every case the loss each period is a constant fraction of whatever remains, not a fixed dollar or unit amount.