Mathematics · Precalculus
Exponential Half-Life Calculator
Calculate half-life from a positive continuous decay-rate magnitude.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- ln(2)÷0.1=6.931471805599452 periods.
- Report Half-life=6.931471805599452, Decay constant=0.1.
Understand Exponential half-life
One idea, three depths
Choose how deeply to explain Exponential half-life
Exponential half-life: Calculate half-life from a positive continuous decay-rate magnitude.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Exponential half-life to answer this question: calculate half-life from a positive continuous decay-rate magnitude? Enter Decay-rate magnitude; the calculator shows Half-life. For example: A 10% continuous decay rate has half-life about 6.93 periods. The answer tells you Half-life.
Age 15Explain it to a 15-year-oldConnect it to the formula
A half-life is the time for exponential decay to multiply a quantity by one half. The rule is t½=ln(2)/k. Its input values are Decay-rate magnitude (% per period), and the main result is Half-life. For example: A 10% continuous decay rate has half-life about 6.93 periods.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated exponential half-life relation over the valid real-number domain stated below. The implemented relation is t½=ln(2)/k, evaluated from Decay-rate magnitude (% per period) to produce Half-life. A half-life is the time for exponential decay to multiply a quantity by one half. Enter the positive magnitude of the decay rate.
Inputs and valid domain
- Decay-rate magnitude must be a finite real number, at least 0 in % per period.
Important boundary: Enter the positive magnitude of the decay rate.
The formula
t½=ln(2)/k
How the calculator works through it
It substitutes Decay-rate magnitude into the formula and exposes every numerical step above. The main output is Half-life, accompanied by Decay constant.
Read the result correctly
The Half-life is the direct answer to “calculate half-life from a positive continuous decay-rate magnitude.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
A 10% continuous decay rate has half-life about 6.93 periods.
Where this model stops being reliable
Enter the positive magnitude of the decay rate.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Exponential half-life works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Exponential half-life uses t½=ln(2)/k. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Functions, domains and ranges
Domain and range language helps you identify which Exponential half-life inputs are valid and how the output behaves.
Review this foundation about 6 min
Optional enrichment
- Exponential growth and decay
Exponential models provide a useful extension when Exponential half-life is applied to multiplicative change.
Review this foundation about 6 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Decay-rate magnitude.
- Evaluate the principal relationship: t½=ln(2)/k.
- Return Half-life and check the domain conditions described above.
Python
from math import *
def exponential_half_life(a) -> float:
return (log(2.0) / (a / 100.0))
assert abs(exponential_half_life(10) - 6.931471805599452) < 1e-6 * max(1.0, abs(6.931471805599452))
C
#include <assert.h>
#include <math.h>
double exponential_half_life(double a) {
return (log(2.0) / (a / 100.0));
}
int main(void) {
const double expected = 6.931471805599452;
const double actual = exponential_half_life(10);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double exponential_half_life(double a) {
return (std::log(2.0) / (a / 100.0));
}
int main() {
constexpr double expected = 6.931471805599452;
const double actual = exponential_half_life(10);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double exponential_half_life(double a)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern log
global exponential_half_life
section .text
exponential_half_life:
push rbp
mov rbp, rsp
sub rsp, 48
movsd [rbp-8], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
call log wrt ..plt
movsd [rbp-24], xmm0
mov rax, 0x4059000000000000
movq xmm0, rax
movsd [rbp-48], xmm0
movsd xmm0, [rbp-8]
divsd xmm0, [rbp-48]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-24]
divsd xmm0, [rbp-40]
movsd [rbp-16], xmm0
movsd xmm0, [rbp-16]
leave
ret
MATLAB
function result = exponential_half_life(a)
result = (log(2.0) / (a / 100.0));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_] := (Log[2.0] / (a / 100.0));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Exponential Half-Life Calculator. MW SysArc Tools. https://math.mwsysarc.com/precalculus/exponential-half-life
MLA 9
MW SysArc. “Exponential Half-Life Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/precalculus/exponential-half-life. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Exponential Half-Life Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/precalculus/exponential-half-life.
Harvard
MW SysArc (2026) ‘Exponential Half-Life Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/precalculus/exponential-half-life (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_exponential_half_life_2026,
author = {{MW SysArc}},
title = {Exponential Half-Life Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/precalculus/exponential-half-life},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Exponential Half-Life Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/precalculus/exponential-half-life
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Exponential half-life do?
Calculate half-life from a positive continuous decay-rate magnitude.
How does the Exponential half-life work?
The calculator applies t½=ln(2)/k. A half-life is the time for exponential decay to multiply a quantity by one half.
What can I learn from the Exponential half-life?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .