Exponential decay is a decrease by the same percent in each equal time period, so the amount drops quickly at first and then more and more slowly. It is modeled by y = a(1 − r)t, where a is the starting amount and r is the decay rate. The amount approaches 0 but never reaches it.
In Spanish
decaimiento exponencial
Also called
exponential decrease, decay factor model, half-life model
Each half-life cuts the amount in half: 80 → 40 → 20 → 10 → 5. The drops get smaller, and the amount never quite reaches 0.
Formal definition and notation
A quantity decays exponentially when it can be written as y = a · bt with a > 0 and decay factor 0 < b < 1. Writing b = 1 − r shows the decay rate r, with 0 < r < 1. Over every interval of equal length the quantity is multiplied by the same factor, and y → 0 as t → ∞. Continuous decay is written y = aekt with k < 0.
Symbol
Read it as
Meaning
1 − r
one minus r
the decay factor: the fraction that remains after each period
a( 12 )th
a times one half to the t over h
half-life form: the amount halves every h time units
ae−kt
a e to the negative k t
continuous decay with rate k > 0
Key facts
The decay factor is between 0 and 1. A 30% decrease per year has factor 1 − 0.30 = 0.7.
Each period removes the same percent, so the amount removed shrinks: 80 to 40 loses 40, but 40 to 20 loses only 20.
The graph falls from left to right and levels off toward the horizontal asymptotey = 0.
Half-life is the time for an amount to fall to half. After n half-lives, ( 12 )n of it is left.
Real examples: a car losing value, a medicine leaving the body, a radioactive sample, a cooling drink approaching room temperature.
Decay is exponential growth run with a factor below 1; y = 50(0.8)t is the same as y = 50(1.25)−t.
Exponential decay formula
Exponential decay formula
y = a(1 − r)t
y
the amount left after t periods
a
the starting amount
r
the percent lost per period, as a decimal
t
the number of periods
A 15% loss leaves 85%, so the factor is 0.85. With a half-life h, use y = a( 12 )th.
Try it: lose 20% each step
Interactive
The machine multiplies by the decay factor 0.8, which is a 20% loss. It has already run 100 and 80. Type the newest output back in and press Run: each drop is smaller than the one before.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Exponential Decay examples, step by step
Example 1: Half-life
Warm-up
A 200 mg dose of a medicine has a half-life of 6 hours. How much is left after 24 hours?
A linear decrease subtracts the same amount each period (minus 50 a year) and hits 0. Exponential decay removes the same percent (minus 10% a year) and only approaches 0.
Decay comes from a factor between 0 and 1, like 0.5t. A negative exponent is one way to write it: 0.5t = 2−t. The exponent is not what makes it decay; the factor is.
Quick check: 3 questions
Question 1 of 3
Which model shows a 40% decrease each year from 500?
Show answer and explanation
Answer: C. Losing 40% keeps 60%, so the factor is 0.6. 0.4 keeps only 40%, 500 − 40t subtracts a fixed amount, and 1.4 is 40% growth.
Question 2 of 3
A 64 g sample has a half-life of 10 days. How much remains after 30 days?
Show answer and explanation
Answer: A. 30 days is 3 half-lives: 64 → 32 → 16 → 8. 16 g stops after 2 half-lives, 21.3 g divides by 3, and 32 g is one half-life.
Question 3 of 3
Which function shows exponential decay?
Show answer and explanation
Answer: D. Decay needs a base between 0 and 1: 0.95. 3(1.05)x grows, 0.5(3)x grows (the 0.5 is only the starting value), and −2x + 10 is a linear decrease.
Exponential Decay: frequently asked questions
What is the formula for exponential decay?
The basic formula is y = a(1 − r)t, where a is the starting amount, r is the percent lost per period as a decimal and t is the number of periods. For half-life problems use y = a( 12 )th, where h is the half-life. Science courses also use y = ae−kt.
How do you tell exponential growth from decay?
Look at the base, or factor, b in y = abx. If b > 1, the function grows; if 0 < b < 1, it decays. The starting value a does not decide it: y = 0.2(5)x is growth and y = 200(0.5)x is decay. In a word problem, “increase” signals growth and “decrease” or “depreciate” signals decay.
Does exponential decay ever reach zero?
No. Each period keeps a fixed fraction of what is there, and a fraction of a positive number is still positive. The values get as close to 0 as you like, so the graph has the horizontal asymptote y = 0, but it never touches it. In real life a model stops being useful once the amount is tiny, like the last few atoms of a sample.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-LE.A.1.c, CCSS HSF-IF.C.8.b
Cite this entry
Effortless Math Team. “Exponential Decay.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/exponential-decay/
Flashcard
TermExponential Decaysame percent lost every period
DefinitionAn amount multiplied by the same factor between 0 and 1 each period: y = a(1 − r)t. Example: 80 halving gives 80, 40, 20, 10.
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