Logarithmic Regression Calculator

Fit a logarithmic model y = a + b ln(x) to your (x, y) data using least squares. See the coefficients a and b, the coefficient of determination R², a scatter plot with the fitted curve, and a table of predicted values.

Input

Enter your (x, y) data points to fit the logarithmic model y = a + b ln(x) by least squares. Only positive x values are supported.

Data points (x, y)

One point per line, with x and y separated by a comma or a space. At least 3 points are required.

Result

Regression equation

y = 2.1673 + 1.8266 ln(x)

Intercept a

2.167285

Coefficient b

1.826627

R²

0.9991

Correlation r

0.9995

Data points

Scatter plot and fitted curve

Data points and predictions

x	Observed y	Predicted y
1	2.1	2.1673
2	3.5	3.4334
3	4.2	4.174
5	5.1	5.1071
8	6	5.9656
13	6.8	6.8525

How it works

The model is y = a + b ln(x), defined only for positive x. Substituting u = ln(x) turns it into the simple linear regression y = a + b u, so least squares gives a and b directly.
The slope is b = Sum of (u − mean u)(y − mean y) divided by Sum of (u − mean u) squared, and the intercept is a = mean y − b times mean u, where mean u and mean y are the averages of u and y.
The coefficient of determination R² equals 1 minus the residual sum of squares divided by the total sum of squares. Values closer to 1 indicate a better fit. The correlation r is the Pearson correlation between u and y, and R² equals r squared.
Enter one point per line with x and y separated by a comma or a space. At least 3 points are required. Points where x is 0 or less, or data where every x is identical, cannot be fitted.

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