Simplify Your Workflow: Search MiniWebtool.
Add Extension
Home Page > Math > Advanced Math Operations > Exponential Decay Calculator

Exponential Decay Calculator

Calculate exponential decay with interactive visualizations, half-life computation, decay constant, and step-by-step solutions. Solve for initial amount, final amount, decay rate, or time.

Exponential Decay Calculator
P(t)
Final Amount
Calculate remaining
P₀
Initial Amount
Find original
r
Decay Rate
Find rate
t
Time
Calculate duration
Quick Examples: Radioactive Decay Find Initial Find Rate Half-Life
P₀ Initial Amount
Starting quantity
P(t) Final Amount
Amount after decay
r Decay Rate
Use decimal (0.05 = 5%)
t Time
Time periods elapsed
Precision
Result accuracy

Embed Exponential Decay Calculator Widget

About Exponential Decay Calculator

Welcome to the Exponential Decay Calculator, a comprehensive tool for solving exponential decay problems with step-by-step solutions and interactive visualizations. Whether you need to calculate the final amount after decay, determine the initial amount, find the decay rate, or compute the time required for decay, this calculator provides accurate results with detailed explanations.

What is Exponential Decay?

Exponential decay describes the decrease of a quantity at a rate proportional to its current value. Unlike linear decay where a fixed amount is lost each period, exponential decay removes a fixed percentage, causing the quantity to decrease more slowly as it gets smaller. This behavior is described by the continuous decay formula:

$$P(t) = P_0 \cdot e^{-rt}$$

Where:

Exponential Decay Calculator Features

Four Calculation Modes

This calculator can solve for any variable in the exponential decay equation:

Additional Calculations

Beyond the main result, the calculator also provides:

Interactive Decay Curve

The calculator generates a visual representation of the decay process, showing how the quantity decreases over time with the calculated point marked on the curve.

Time-Series Table

A detailed table shows the decay progression at regular time intervals, including amount remaining, amount decayed, and percentage remaining at each point.

How to Use This Calculator

  1. Select what to solve for: Choose what variable you want to calculate (Final Amount, Initial Amount, Decay Rate, or Time). The calculator will display the required input fields.
  2. Enter known values: Input the values you know. For decay rate, use decimal format (0.05 for 5%). All values must be positive.
  3. Select precision: Choose the number of decimal places for your result (4 to 10 decimal places).
  4. Calculate: Click the Calculate button to see your result with step-by-step solution, decay curve, and time-series table.
  5. Analyze results: Review the decay visualization and additional metrics like half-life and decay constant.

Understanding Half-Life

Half-life is the time required for a quantity to reduce to half its initial value. It is calculated using:

$$t_{1/2} = \frac{\ln(2)}{r} \approx \frac{0.693}{r}$$
Key insight: Half-life is constant regardless of the initial amount. After one half-life, 50% remains. After two half-lives, 25% remains. After three half-lives, 12.5% remains, and so on.

Decay Rate vs. Decay Constant

In the continuous decay formula P(t) = P₀e^(-rt), the decay rate r and decay constant λ (lambda) are equivalent. They represent how quickly the quantity decreases:

Real-World Applications

Radioactive Decay

Radioactive isotopes decay at rates characterized by their half-lives. Carbon-14 has a half-life of about 5,730 years, making it useful for dating organic materials up to about 50,000 years old.

Example: If you have 100g of a radioactive isotope with a half-life of 10 years, after 30 years (3 half-lives), you would have 100 × (1/2)³ = 12.5g remaining.

Drug Metabolism (Pharmacokinetics)

Drugs are eliminated from the body through exponential decay. The elimination half-life determines how often a drug needs to be administered to maintain therapeutic levels.

Asset Depreciation

Some financial models use exponential decay to model how assets like vehicles and electronics lose value over time.

Population Decline

Declining populations often follow exponential decay patterns when the death rate exceeds the birth rate by a consistent proportion.

Cooling and Heating (Newton's Law)

The temperature difference between an object and its environment decreases exponentially over time according to Newton's Law of Cooling.

Electrical Circuits

Capacitors discharge through resistors following exponential decay, characterized by the RC time constant.

Related Formulas

Solving for Different Variables

The exponential decay formula can be rearranged to solve for any variable:

$$P_0 = P(t) \cdot e^{rt}$$ $$r = -\frac{\ln(P(t)/P_0)}{t}$$ $$t = -\frac{\ln(P(t)/P_0)}{r}$$

Discrete vs. Continuous Decay

While this calculator uses continuous exponential decay (base e), discrete decay uses a different formula:

$$P(t) = P_0 \cdot (1 - r)^t$$

For continuous decay: P(t) = P₀e^(-rt)
For discrete decay: P(t) = P₀(1-r)^t

Frequently Asked Questions

What is exponential decay?

Exponential decay describes the decrease of a quantity at a rate proportional to its current value. It follows the formula P(t) = P₀ × e^(-rt), where P₀ is the initial amount, r is the decay rate, and t is time. Common examples include radioactive decay, drug metabolism, and depreciation.

How do I calculate final amount after decay?

To calculate the final amount, use the formula P(t) = P₀ × e^(-rt). Enter your initial amount P₀, decay rate r, and time t. The calculator multiplies the initial amount by e raised to the power of negative r times t to give you the remaining amount.

What is half-life in exponential decay?

Half-life is the time required for a quantity to reduce to half its initial value. It is calculated as t½ = ln(2) / r, where r is the decay rate. Half-life is constant regardless of the initial amount and is commonly used in radioactive decay and pharmacology.

What is the difference between decay rate and decay constant?

In continuous exponential decay using P(t) = P₀ × e^(-rt), the decay rate r and decay constant λ (lambda) are the same value. They represent how quickly the quantity decreases. A higher value means faster decay. The decay constant is often denoted by the Greek letter lambda.

What are real-world applications of exponential decay?

Exponential decay models many natural and financial phenomena including: radioactive decay of isotopes, drug concentration in the bloodstream over time, depreciation of assets, population decline, cooling of objects (Newton's law of cooling), discharge of capacitors in electronics, and decay of sound intensity.

How do I calculate decay rate from initial and final amounts?

Use the formula r = -ln(P(t)/P₀) / t. Divide the final amount by the initial amount, take the natural logarithm, divide by time, and negate the result. This gives you the decay rate per time unit.

What happens if my decay rate is negative?

A negative decay rate actually represents exponential growth, not decay. For true decay, the rate must be positive, meaning the quantity decreases over time. Use a positive decay rate for this calculator.

Additional Resources

For further learning about exponential decay:

Reference this content, page, or tool as:

"Exponential Decay Calculator" at https://MiniWebtool.com/exponential-decay-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 12, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Related MiniWebtools:

Exponential Growth Calculator (High Precision)
Exponential Integral Calculator
Half Life Calculator

Advanced Math Operations:

Top & Updated:

Random PickerRandom Name PickerRelative Standard Deviation CalculatorMAC Address GeneratorFPS ConverterMAC Address LookupSort NumbersFeet and Inches to Cm ConverterRemove SpacesWord to Phone Number ConverterLine CounterRandom Quote GeneratorPercent Off CalculatorSum CalculatorBitwise CalculatorBatting Average CalculatorNumber of Digits CalculatorRoman Numerals ConverterFacebook User ID LookupLog Base 10 CalculatorSHA256 Hash GeneratorSalary Conversion CalculatorPhone Number ExtractorSquare Root (√) CalculatorSaturn Return CalculatorVideo to Image ExtractorDecimal to BCD ConverterSun, Moon & Rising Sign Calculator 🌞🌙✨Remove AccentERA CalculatorAI Punctuation AdderMP3 LooperCm to Feet and Inches ConverterBinary to Gray Code ConverterBCD to Decimal ConverterCompound Growth CalculatorRemove Audio from VideoQuotient and Remainder CalculatorRemove Lines Containing...Percent Growth Rate CalculatorProportion CalculatorRandom Birthday GeneratorTime Duration CalculatorFirst n Digits of PiAI ParaphraserOutlier CalculatorDay of Year CalendarGray Code to Binary ConverterLove Compatibility CalculatorText FormatterModulo CalculatorPercentage Increase CalculatorList of Prime NumbersCompare Two StringsMaster Number CalculatorNumber to Word ConverterMerge VideosBinary to BCD ConverterIP Address to Hex ConverterLeap Years ListAudio SplitterPER Calculator📅 Date CalculatorHex to BCD ConverterAdd or Replace Audio in VideoScientific Notation to Decimal ConverterOn Base Percentage CalculatorRemove Leading Trailing SpacesFraction CalculatorURL ExtractorHypotenuse CalculatorNatural Log CalculatorNumber ExtractorArctan2 CalculatorBCD to Binary ConverterRandom Object GeneratorMedian CalculatorRatio to Percentage CalculatorPythagorean Theorem CalculatorVideo CompressorAverage Deviation CalculatorRandom Group GeneratorAdd Prefix and Suffix to TextLog Base 2 CalculatorSHA512 Hash GeneratorASCII Art GeneratorBreak Line by CharactersBonus CalculatorInvisible Text GeneratorAdjust Video SpeedOctal ConverterSlugging Percentage CalculatorStandard Error CalculatorName Number CalculatorConvolution CalculatorDay of the Year Calculator - What Day of the Year Is It Today?Multiplication CalculatorFirst n Digits of eAmortization CalculatorOctal CalculatorEmail ExtractorOPS CalculatorWHIP CalculatorSort Lines AlphabeticallyLottery Number GeneratorBCD to Hex ConverterList RandomizerAverage CalculatorPVIFA CalculatorReverse VideoWAR CalculatorActual Cash Value CalculatorAngel Number CalculatorRoot Mean Square CalculatorSum of Positive Integers CalculatorSHA3-256 Hash GeneratorAI Sentence ExpanderLbs to Kg ConverterHex to Decimal ConverterMAC Address AnalyzerRandom String GeneratorMarkup CalculatorPVIF CalculatorDecimal to Hex ConverterList of Fibonacci NumbersRandom Number PickerBlood Donation Time CalculatorAntilog CalculatorField Goal Percentage CalculatorRandom Line PickerAPY to APR CalculatorSort Text By LengthOne Rep Max (1RM) CalculatorName RandomizerRandom Animal GeneratorSquare Numbers ListFixed Deposit CalculatorPPM to Percent ConverterCube Numbers ListCaesar Cipher ToolMorse Code GeneratorPercent to PPM ConverterSRT to TXT Converterkg to lbs ConverterXML ValidatorArithmetic Mean CalculatorBoard Foot Calculator 🪵Squareness Checker (3-4-5 Rule) 📐Miter Angle Calculator 📐Slope and Grade Calculator 📐Sine CalculatorCosine CalculatorHigh-Precision Tangent CalculatorArcsin CalculatorArccos (Inverse Cosine) CalculatorArctan CalculatorCosecant/Secant/Cotangent CalculatorHyperbolic Functions CalculatorRight Triangle CalculatorGeneral Triangle SolverLaw of Sines CalculatorLaw of Cosines CalculatorDMS to Decimal Degrees ConverterDecimal Degrees to DMS ConverterHigh-Precision Polar to Cartesian Coordinates ConverterHigh-Precision Cartesian to Polar Coordinates ConverterInteractive Unit Circle VisualizerTrigonometric Identities CalculatorTrigonometric Function GrapherAlgebraic Expression SimplifierRadical SimplifierRational Expression CalculatorPolynomial Factoring CalculatorExpand Polynomials CalculatorPolynomial Long Division CalculatorSynthetic Division CalculatorLinear Equation SolverAbsolute Value Equation SolverRadical Equation SolverSystem of Linear Equations SolverInequality SolverAbsolute Value Inequality SolverSystem of Inequalities GrapherFunction GrapherDomain and Range CalculatorInverse Function CalculatorFunction Composition CalculatorVertex and Axis of Symmetry CalculatorX and Y Intercept CalculatorHTML Entity Encoder/DecoderURL Slug GeneratorJSON String Escape/UnescapeMarkdown Table GeneratorSmart Quotes RemoverDaily Time Savings CalculatorLorem Ipsum GeneratorReadability Score CalculatorKeyword Density CheckerBionic Reading ConverterText to Speech ReaderFancy Text GeneratorZalgo Text GeneratorUpside Down Text GeneratorList Difference CalculatorText Column ExtractorText to SQL List ConverterInvisible Character RemoverStock Average CalculatorPosition Size CalculatorMargin Call CalculatorShort Selling Profit CalculatorRisk of Ruin CalculatorBlack-Scholes Option Pricing CalculatorOption Greeks CalculatorImplied Volatility CalculatorOptions Profit CalculatorImpermanent Loss CalculatorCrypto Arbitrage CalculatorSatoshi to USD ConverterCrypto Leverage CalculatorPivot Point CalculatorFibonacci Extension CalculatorStop Loss & Take Profit Calculator