Calculator Inputs
Example Data Table
| x | y | Context |
|---|---|---|
| 1 | 2.1 | Low input response |
| 2 | 3.0 | Early growth |
| 3 | 3.6 | Moderate increase |
| 4 | 4.2 | Diminishing gains |
| 5 | 4.7 | Flattening trend |
- Learning curves where improvements slow over time.
- Returns that diminish as input grows.
- Processes that respond quickly at first, then level off.
Formula Used
a = ȳ − b·ū, where u = log(x)
- R² = 1 − SSE/SST (when SST > 0)
- SSE = Σ(y − ŷ)²
- RMSE = √(SSE/n)
- MAE = Σ|y − ŷ| / n
How to Use This Calculator
- Enter your paired data as x, y values, one line per point.
- Choose the separator style and preferred log base.
- Optionally enter an x value to predict ŷ from the fitted curve.
- Press Submit to compute coefficients and diagnostics.
- Use Download CSV or Download PDF after results appear.
Practical meaning of a logarithmic fit
A logarithmic regression is useful when increases happen fast at the beginning and then slow down. This pattern appears in learning curves, diminishing returns, and saturation effects. The fitted curve summarizes how much y changes when x grows multiplicatively. In this calculator, the transformed variable is u = log(x), so the model behaves like a straight line in u.
Interpreting the coefficients a and b
The intercept a is the estimated value when log(x) equals zero. For ln, that reference point is x = 1. The slope b measures the expected change in y for a one-unit increase in log(x). With ln, a one-unit increase corresponds to multiplying x by e. With log10, it corresponds to multiplying x by 10. Larger |b| means a stronger diminishing-returns effect.
Diagnostics that indicate fit quality
R² compares the explained variation to total variation. Values closer to 1 generally indicate a tighter fit, but always check residuals. RMSE reports a typical error magnitude in y-units, while MAE provides an average absolute deviation that is less sensitive to large outliers. SSE aggregates squared errors and is useful for comparing fits on the same dataset.
Residual patterns and data readiness
Residuals should look randomly scattered around zero. If residuals trend upward or downward with x, the relationship may need a different curve. If residuals increase in spread as x grows, consider scaling, segmenting, or reviewing measurement noise. Because log(x) is required, all x values must be strictly positive. If your dataset includes zeros, shift or filter the input using a defensible rule.
Export workflow for reporting
After submitting points, the results panel provides a fitted equation, prediction option, and a table with log(x), ŷ, and residuals. Use CSV export to move rows into spreadsheets for charts or audits. Use PDF export to capture the model, summary metrics, and a preview of the first rows for documentation. For consistent reporting, keep the same log base across comparisons and record the exact data points used for the run.
FAQs
1) What does logarithmic regression model?
It models relationships where y increases quickly for small x, then grows more slowly as x gets larger. It is common in learning, saturation,
and diminishing-returns processes.
2) Why must x be greater than zero?
Logarithms are only defined for positive values in real-number analysis. Any zero or negative x prevents computing log(x), so the model cannot be fitted.
3) Should I choose ln or log10?
Either works; they differ only by a constant scale factor. Choose ln for scientific and continuous-growth contexts, or log10 for decade-based interpretation. Keep the same base when comparing results.
4) How do I read R², RMSE, and MAE together?
Use R² for relative fit, and RMSE/MAE for error size in y-units. A high R² with large RMSE suggests the curve captures the trend but predictions may still be noisy.
5) Can I predict y for a new x value?
Yes. Enter a positive x in the prediction field. The calculator evaluates the fitted equation to return ŷ. Predictions outside the data range should be treated cautiously.
6) What if residuals show a clear pattern?
A pattern usually means the model form is mismatched. Try a different curve, check for outliers, or split the data into regimes. Residual randomness is a key sign of adequacy.