Half-life is the time it takes for a quantity undergoing exponential decay to reduce to exactly half of its current amount. It's constant for a given substance or process, regardless of the starting amount.

What is the half-life formula?

N = N₀ × (1/2)^(t / half-life), where N₀ is the initial amount, N is the remaining amount after time t, and half-life is the time for the quantity to halve.

How do you solve for elapsed time (t)?

t = half-life × [ln(N / N₀) / ln(0.5)]. This requires N and N₀ to both be positive, since you can't take the logarithm of a negative number or zero.

How do you solve for the half-life itself?

half-life = [t × ln(0.5)] / ln(N / N₀). This is useful when you've measured how much of a substance remains after a known time and want to determine its decay rate.

What happens if I leave more than one field blank?

The calculator needs exactly three known values to solve for the fourth. If you leave more than one field blank, it will show an error asking you to leave exactly one blank.

Can the remaining amount (N) be greater than the initial amount (N₀)?

No, not in a decay scenario — N should always be less than or equal to N₀. If you enter a remaining amount larger than the initial amount while solving for time or half-life, the result won't make physical sense.

Why do I need N₀ and N to be positive when solving for time or half-life?

Solving for t or half-life involves taking ln(N/N₀), and logarithms are undefined for zero or negative arguments — both amounts represent physical quantities that must be positive.

Where is half-life used in real life?

It's used in radiometric dating (like carbon-14 dating of archaeological samples), pharmacology (how long a drug stays active in the body), and nuclear physics and safety (how long radioactive material remains hazardous).

Half-Life Calculator — Solve for Any Missing Value

Our Half-Life Calculator solves N = N₀ × (1/2)^(t / half-life) for whichever value you don't know. Enter the three values you do know — any combination of initial amount (N₀), remaining amount (N), elapsed time (t), and half-life — and leave the fourth blank; the calculator isolates and solves for it algebraically, showing every step. Useful for radioactive dating, medication dosing schedules, and any decay problem where you need to work backward from a known half-life.

Quick Answer

The half-life formula is N = N₀(1/2)^(t/half-life). Enter any three of initial amount, remaining amount, time, and half-life, leave the fourth blank, and this calculator solves for it with the full algebra shown.

N = N₀ × (1/2)^(t / half-life) — leave one field blank to solve for it

Initial amount (N₀)

Remaining amount (N)

Time (t)

Half-life

Enter three values and leave one blank, then click Calculate.

How to Use the Half-Life Calculator — Solve N = N₀(1/2)^(t/half-life) Online

1
Enter the three values you know: initial amount, remaining amount, elapsed time, or half-life.
2
Leave the field you want to solve for blank.
3
Click 'Calculate' to solve the half-life equation for the missing value.
4
Review the step-by-step algebra to see exactly how the result was derived.

Why Use Half-Life Calculator — Solve N = N₀(1/2)^(t/half-life) Online?

Half-life problems come in more than one direction: sometimes you know the half-life and need to project how much remains after a given time, but just as often you know the initial and remaining amounts and need to work backward to find either the elapsed time or the half-life itself — like dating a radioactive sample or determining a drug's half-life from blood-test measurements. Rearranging N = N₀(1/2)^(t/half-life) to solve for t or half-life requires logarithms, which is easy to set up with the wrong base or sign. This calculator solves for any of the four variables and shows the full algebra.

Frequently Asked Questions

Related Tools

Exponential Decay Calculator — A = A₀e^(-rt) Online

Calculate exponential decay using A = A₀e^(-rt) — enter initial amount, decay rate, and time to find the remaining amount and half-life. Free calculator with step-by-step solution.

Exponential Growth Calculator — A = A₀e^(rt) Online

Calculate exponential growth using A = A₀e^(rt) — enter initial amount, growth rate, and time to find the final amount and doubling time. Free calculator with step-by-step solution.

Proportion Calculator — Solve a/b = c/d Online

Solve proportion problems instantly — enter three known values in a/b = c/d and find the missing fourth value. Free calculator with step-by-step cross-multiplication.