How this instrument works
Exponential growth or decay can be written two equivalent ways: the discrete form, y = a·bˣ, where b is the growth factor applied once per whole period (b=2 for doubling, b=0.5 for halving), or the continuous form, y = a·e^(kx), built on the natural constant e with a continuous rate k. Converting from the discrete base to the matching continuous rate is a single logarithm: k = ln(b), since a·bˣ and a·e^(ln(b)·x) are the identical function, just written with a different base raised to a different-looking exponent.
The continuous form is often more convenient in calculus and in modeling continuously compounding processes (population growth, radioactive decay, continuously compounded interest), where working directly with e simplifies differentiation and integration — the derivative of e^(kx) is simply k·e^(kx), a much cleaner relationship than differentiating bˣ directly for an arbitrary base b.
The sign of k carries the same meaning the size of b did: a growth base b greater than 1 (genuine growth) converts to a positive k, a base between 0 and 1 (decay) converts to a negative k, and a base of exactly 1 (no change at all) converts to k=0 — the two forms always agree on which direction the quantity is actually moving.
- Enter the discrete growth base (b, from y = a·bˣ) into the Growth base field.
- Read Continuous rate: the sheet applies the natural logarithm directly.
- Try a base between 0 and 1 to see the resulting rate come out negative, matching decay rather than growth.
Worked example — a doubling base of 2
A quantity doubling every period, y = a·2ˣ, converts to continuous exponential form as y = a·e^(0.693x) — check: ln(2) ≈ 0.693, and e^0.693 ≈ 2, confirming the two forms describe the identical growth curve.
A growth base of exactly 1 (no change at all) converts to a continuous rate of exactly 0, since ln(1)=0 — a perfectly flat, unchanging quantity either way it's written. A growth base of e itself converts to a continuous rate of exactly 1, since e already IS its own natural continuous form, needing no real conversion at all.
Questions
How do you convert a discrete growth rate to a continuous one?
Take the natural logarithm of the discrete base: k = ln(b). This gives the continuous rate k such that y = a·bˣ and y = a·e^(kx) describe the identical curve.
Why use the continuous e-based form at all?
It simplifies calculus significantly — the derivative of e^(kx) is simply k·e^(kx), a much cleaner relationship than differentiating an arbitrary base bˣ directly, which is exactly why continuous growth and decay models in science are usually built on e.
What does a negative continuous rate mean?
Decay rather than growth — it corresponds to a discrete base b between 0 and 1, since ln(b) is negative for any b in that range, matching a quantity that shrinks rather than grows over time.
What if the growth base is 1?
The continuous rate comes out to exactly 0, since ln(1)=0 — a quantity that neither grows nor shrinks, staying perfectly flat, converts consistently to a continuous rate of zero as well.
Can the growth base be negative or zero?
No — the natural logarithm is only defined for positive numbers, and a discrete growth base of zero or negative wouldn't describe a sensible ongoing growth or decay process in the first place.