Formula Used
Exponential interpolation treats the dependent value as changing by a constant percentage per unit of x. To support shifted datasets, we use an optional offset.
t = (x − x₁) / (x₂ − x₁)
y(x) = y₁′ · exp( ln(y₂′/y₁′) · t ) + offset
The continuous growth rate is k = ln(y₂′/y₁′) / (x₂ − x₁). The multiplicative factor per +1 step is r = e^k.
How to Use This Calculator
- Enter two data points: (x₁, y₁) and (x₂, y₂).
- Type the target x where you want the estimate.
- If any y-value is zero or negative, add a small offset.
- Choose the desired precision and press Calculate.
- Use the download buttons to export a CSV or a PDF summary.
Example Data Table
| Case | x₁ | y₁ | x₂ | y₂ | Target x | Offset | Estimated y(x) |
|---|---|---|---|---|---|---|---|
| Growth | 0 | 100 | 10 | 200 | 6 | 0 | 151.571656 |
| Decay | 0 | 80 | 5 | 20 | 3 | 0 | 31.748021 |
| Shifted | 1 | 5 | 4 | 9 | 2.5 | 4 | 7.000000 |
Example values are rounded for display. Your precision setting controls output rounding.
Article: Exponential Interpolation in Practice
1) What exponential interpolation means
Exponential interpolation estimates a value between two measurements by assuming a constant percentage change. If a quantity doubles over an interval, it is treated like compounding rather than adding a fixed amount. This matches many real processes such as population growth, radioactive decay, and compound interest.
2) Why the log step matters
The calculator works in “log space” using the ratio y₂′/y₁′ and the natural log ln(ratio). This converts multiplicative change into additive change, which is stable for large ranges. For example, moving from 100 to 200 is a ratio of 2, while 100 to 400 is a ratio of 4.
3) The two-point growth model
With points (x₁, y₁) and (x₂, y₂), the fitted curve is y(x)=y₁′·exp(ln(y₂′/y₁′)·t)+offset, where t=(x−x₁)/(x₂−x₁). When x is halfway, t=0.5 and the estimate becomes the geometric mean in the shifted space.
4) Reading the rate outputs
The continuous rate k equals ln(y₂′/y₁′)/(x₂−x₁). If k=0.0693, the value grows about 7.18% per +1 x because r=e^k≈1.0718. A negative k indicates decay, so r is below 1 and the percent per step is negative.
5) Using the offset for shifted data
Exponential models require positive values, so the offset creates y′=y−offset. This is useful for baselines such as “signal above noise” or “production above idle level.” Keep the offset smaller than both y-values so y′ remains positive at each anchor point.
6) Interpolation versus extrapolation
When t stays between 0 and 1, you are interpolating between known points. If t<0 or t>1, you are extrapolating beyond the measured range and uncertainty rises quickly. A small error in the ratio can become a large error when the curve is extended far.
7) Real-world uses and quick checks
Common uses include forecasting sensor drift, scaling lab concentrations, estimating battery discharge, and filling missing values in time series. A quick sanity check is to compare with linear interpolation: if both answers are close, the data is near-linear; if they differ, the process may be truly multiplicative. In the Growth example (100→200 over 0→10), x=6 gives about 151.57. In the Decay example (80→20 over 0→5), x=3 gives about 31.75. With offset=4, the shifted case 5→9 becomes 1→5, giving 7 at x=2.5.
FAQs
1) What is the difference between linear and exponential interpolation?
Linear interpolation adds a constant amount per step. Exponential interpolation applies a constant percentage change, so it behaves like compounding. Use exponential when ratios stay more consistent than differences.
2) Why must (y − offset) be positive?
The method uses ln(y₂′/y₁′). Logs require positive inputs, so both y′ values must be above zero. If your data has a baseline, set an offset slightly below the smallest y.
3) What does t represent?
t tells where the target x sits between x₁ and x₂. t=0 returns y₁, t=1 returns y₂, and t=0.5 is the midpoint estimate. Values outside 0–1 indicate extrapolation.
4) What do k and r mean?
k is the continuous growth rate per x unit. r=e^k is the multiplier for a +1 change in x. If r=1.05, the value increases about 5% per step; if r=0.95, it decreases about 5%.
5) Can I use this calculator for time-based forecasting?
Yes, if x represents time and the process approximately follows compounding or decay. Keep forecasts close to your measured interval, and watch for seasonality or regime changes that break a constant-rate assumption.
6) How should I choose decimal precision?
Use 2–4 decimals for quick reporting and 6–10 for engineering or scientific work. Precision only changes rounding of displayed results; it does not change the underlying calculation used for CSV/PDF exports.