LOGARITHMIC CURVE FIT

Logarithmic Regression Calculator

Fit a logarithmic curve ŷ = a + b·ln(x) to paired X and Y observations. The predictor X must be positive. The result includes the equation, graph, R², RMSE and residuals.

Fit Y against ln(X)Positive X requiredBrowser-based

YOUR DATA

Enter paired observations

Paste or upload numeric X and Y columns. All X values must be greater than zero. Use a period as the decimal mark. Up to 5 MB and 5,000 observations are accepted.

Upload a CSV or TSV file

The file is read locally in your browser.

Or paste data

At least two complete pairs with positive X values are needed.

X variable

Y variable

First row contains column namesTurn this off when the first line is data.

YOUR RESULTS

Logarithmic fit

Fitted equation—ŷ = a + b·ln(x)

R-squared (R²)—Proportion of observed variation explained by the fit

RMSE—Typical prediction error in Y units

Parameter a—Intercept

Parameter b—Coefficient of ln(x)

Fitting method—Ordinary least squares on ln(x)

Observed values and fitted curve

Each point is one input observation.

Observed Fitted

Residuals by observation

Residual = observed Y − predicted Y.

Row	Observed Y	Predicted Y	Residual

How to use this calculator

Paste paired X and Y values, or load the worked example.
Choose the X and Y columns. Every X value must be strictly positive because ln(x) is undefined at zero and for negative numbers.
Review the fitted equation, curve, R², RMSE and residuals.
Download a JSON summary that records the method and any exclusions.

Formula and calculation method

The model is ŷ = a + b·ln(x). The calculator transforms each predictor to w = ln(x) and fits the straight line ŷ = a + bw by ordinary least squares. The slope b and intercept a are displayed directly in the original equation.

R² = 1 − SSE/SST and RMSE = √(SSE/n), both computed from the original-scale predictions and observed Y values.

Worked example: X = [1, e, e²] and Y = [0, 1, 2]. Then ln(X) = [0, 1, 2], so the least-squares line through these points is ŷ = 0 + 1·ln(x), giving a = 0 and b = 1.

Assumptions and limitations

Every observed X must be a finite positive number. Zero or negative X values are rejected because ln(x) is undefined.
At least two complete paired observations with distinct positive X values are required.
The fitting criterion is squared error in Y after log-transforming X. It does not test whether a logarithmic relationship is appropriate or whether errors are independent.
R² and RMSE describe fit to these input observations only; they do not prove causation or predict new data.
Extrapolation below zero or far outside the observed X range is not meaningful for this model.
The current input limit is 5,000 observations and 5 MB per file.

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