Exponential Decay Calculator
Calculate exponential decay using P(t) = P₀·e^(-rt). Solve for initial amount, final amount, decay rate, or time with half-life and step-by-step results.
What Is Exponential Decay?
Exponential decay models quantities that shrink by a fixed percentage of their current value each period. Radioactive decay, drug concentration in blood, and capacitor discharge all follow this pattern.
Continuous Decay Formula
$$P(t) = P_0 \cdot e^{-rt}$$Here $P_0$ is the initial amount, $r$ is the decay rate (decay constant), and $t$ is elapsed time. The half-life $t_{1/2}$ is the time needed for half the quantity to remain: $t_{1/2} = \ln(2) / r$.
How to Use This Calculator
- Choose which variable to solve: final amount, initial amount, decay rate, or time.
- Enter the three known values. Use decimal rate (0.05 = 5% per period).
- Results update instantly with half-life and step-by-step algebra.
Compare with Exponential Growth Calculator for increasing quantities, or the Half Life Calculator for nuclear decay problems.
Frequently Asked Questions
What is the difference between exponential and linear decay?
Linear decay removes the same absolute amount each period. Exponential decay removes a fixed percentage, so the absolute loss slows as the quantity shrinks.
Can decay rate be entered as a percentage?
Yes. Enter 5 for 5% per period, or 0.05 directly. Values greater than 1 are treated as percentages and divided by 100.
How is half-life related to the decay constant?
Half-life and decay rate are inverses through $t_{1/2} = \ln(2) / r$. A larger $r$ means faster decay and a shorter half-life.
When should I use $e^{-rt}$ instead of $(1-r)^t$?
Use $e^{-rt}$ for continuous decay (calculus models). Use $(1-r)^t$ for discrete compounding per fixed period. This tool uses the continuous form.
Related Tools
Present Value Factor Table
Cups to Liters Converter
Gallons to Cups Converter
Decrement Clock Time
Spindle Speed Calculator
Common Factors Calculator
TypeScript To Flow
Bikini Calculator
Door Header Size Calculator
Atom Calculator
Mississippi Paycheck Calculator
Language Detector
Pascal Triangle Generator
YAML Validator
Random Year Generator
Invisible Text Generator
Finance Calculator
Draw Analog Clock
Expanded Form Calculator
New Jersey Paycheck Calculator