Pick what you need to solve for — the other two values feed into the exponential decay formula N(t) = N0 × 0.5^(t / half-life).
Half-life is the time it takes for a quantity undergoing exponential decay to fall to exactly half its starting amount. It's a fixed property of the decaying substance or process, no matter how much you start with — after one more half-life passes, whatever remains gets cut in half again.
N(t) = N0 × 0.5^(t / half-life), where N0 is the starting amount, t is the elapsed time, and half-life is the time it takes for the quantity to halve. The exponent t/half-life simply counts how many half-lives have gone by, and each one multiplies what's left by 0.5.
No — exponential decay with a well-defined half-life shows up anywhere a quantity shrinks by the same proportion over equal time intervals. Besides radioactive isotope decay, the same math describes how the body eliminates many drugs and medications over time, and capacitor discharge in electronics. Only the meaning of the quantity and the unit of time change; the formula itself stays identical.
Yes — use the "Solve for" toggle above the inputs. Remaining Amount needs the initial amount, half-life, and elapsed time. Half-Life needs the initial amount, remaining amount, and elapsed time, solved as half-life = t × ln(0.5) / ln(remaining/initial). Elapsed Time needs the initial amount, remaining amount, and half-life, solved as t = half-life × ln(remaining/initial) / ln(0.5). All three modes are the same decay formula, just rearranged algebraically for the unknown you need.
Worked example: starting with 100 units at a half-life of 10 (days, years — any consistent unit), after 20 units of time (2 half-lives): remaining = 100 × 0.5^(20/10) = 100 × 0.25 = 25 units. Working backward, an initial amount of 100 that decays to 25 after 20 units of time implies a half-life of exactly 10, confirming the three modes agree with each other.