Updated Regression & Curve Fitting

Logarithmic Regression Calculator

Fit a logarithmic regression model of the form \(y = a + b \ln(x)\) from your data. Enter paired \(x\) and \(y\) values, get the fitted equation, correlation, \(R^2\), standard error, residuals table and predictions for new \(x\) values with \(x > 0\).

Model: \(y = a + b \ln(x)\) Correlation & \(R^2\) Standard Error & Residuals Predictions for New x

Fit y = a + b ln(x) From Data

This Logarithmic Regression Calculator fits the model \[ y = a + b \ln(x), \] where \(x > 0\). The calculator transforms \(x\) using the natural logarithm, runs a simple linear regression of \(y\) on \(z = \ln(x)\), and reports the fitted coefficients \(a\) and \(b\), the regression equation, correlation \(r\), coefficient of determination \(R^2\), standard error of estimate and a residuals table. You can also enter a new \(x\) value to obtain the predicted \(y\).

Make sure all \(x\)-values are strictly positive, since \(\ln(x)\) is only defined for \(x > 0\). The calculator requires at least 3 valid data points to fit the model.

Enter \(x\) and \(y\) data as comma, space or line-break separated lists. The two lists must have the same length, with all \(x > 0\).

Enter a positive value of \(x\) to compute the prediction \(\hat y(x) = a + b \ln(x)\).

Logarithmic Regression Calculator – Model, Formulas and Interpretation

The Logarithmic Regression Calculator on MyTimeCalculator fits models of the form \[ y = a + b \ln(x), \] where \(x > 0\). This type of model is useful when the response \(y\) changes quickly for small values of \(x\) and more slowly as \(x\) becomes larger, creating a diminishing-returns pattern. The calculator performs the underlying calculations automatically and presents the fitted equation, correlation, \(R^2\), standard error and residuals.

Instead of working directly with \(x\), the method transforms \(x\) using the natural logarithm. Defining \[ z_i = \ln(x_i), \] the model becomes a simple linear regression in the transformed predictor: \[ y_i = a + b z_i + \varepsilon_i. \] Standard linear regression formulas can then be applied to the pairs \((z_i, y_i)\).

1. Formulas for a and b in Logarithmic Regression

Suppose you have \(n\) data points \((x_1, y_1), \dots, (x_n, y_n)\) with all \(x_i > 0\). Define \(z_i = \ln(x_i)\) and the following sums:

\[ S_z = \sum_{i=1}^{n} z_i, \quad S_y = \sum_{i=1}^{n} y_i, \quad S_{zz} = \sum_{i=1}^{n} z_i^2, \quad S_{yy} = \sum_{i=1}^{n} y_i^2, \quad S_{zy} = \sum_{i=1}^{n} z_i y_i. \]

The slope \(b\) and intercept \(a\) of the logarithmic regression line \(\,y = a + b \ln(x)\,\) are then computed as:

\[ b = \frac{n S_{zy} - S_z S_y} {n S_{zz} - S_z^2}, \qquad a = \bar y - b \,\bar z, \]

where \[ \bar y = \frac{S_y}{n}, \quad \bar z = \frac{S_z}{n}. \] These are exactly the same formulas as for ordinary linear regression, but applied to the transformed predictor \(z = \ln(x)\).

2. Fitted Values, R² and Standard Error

Once \(a\) and \(b\) are known, the fitted value at each \(x_i\) is

\[ \hat y_i = a + b \ln(x_i). \]

The calculator computes the residuals \(e_i = y_i - \hat y_i\) and the sums of squares:

\[ \text{SS}_{\text{res}} = \sum_{i=1}^{n} (y_i - \hat y_i)^2, \qquad \text{SS}_{\text{tot}} = \sum_{i=1}^{n} (y_i - \bar y)^2. \]

The coefficient of determination \(R^2\) is defined as

\[ R^2 = 1 - \frac{\text{SS}_{\text{res}}}{\text{SS}_{\text{tot}}}, \]

which measures the proportion of the variability in \(y\) explained by the logarithmic model. The standard error of estimate is

\[ s_e = \sqrt{\frac{\text{SS}_{\text{res}}}{n - 2}}, \]

where \(n - 2\) is the residual degrees of freedom for a two-parameter model with \(a\) and \(b\).

3. Correlation Between y and ln(x)

The calculator also reports the correlation between \(y\) and \(z = \ln(x)\). Using the same sums as above, the correlation coefficient is

\[ r = \frac{n S_{zy} - S_z S_y} {\sqrt{\bigl(n S_{zz} - S_z^2\bigr)\bigl(n S_{yy} - S_y^2\bigr)}}. \]

For a simple regression with one predictor, the squared correlation \(r^2\) matches the \(R^2\) value computed from sums of squares, up to small rounding differences.

4. Predictions from the Logarithmic Model

To predict \(y\) at a new input \(x_\ast > 0\), you use the fitted equation

\[ \hat y(x_\ast) = a + b \ln(x_\ast). \]

The calculator lets you enter a positive value of \(x_\ast\) in the prediction field and returns the corresponding \(\hat y(x_\ast)\). This is useful for interpolation within the range of your existing data or, with caution, for limited extrapolation beyond it.

5. How to Use the Logarithmic Regression Calculator

  1. Prepare your data: collect paired observations \((x_i, y_i)\) where each \(x_i > 0\). Enter the \(x\)-values and \(y\)-values in the two text areas using commas, spaces or line breaks.
  2. Check the data length: make sure both lists have the same number of values and that you have at least three valid data points. The calculator checks for mismatched or invalid inputs.
  3. Run the regression: click the calculate button. The tool computes the sums, transforms \(x\) to \(z = \ln(x)\) internally and fits the model \(y = a + b \ln(x)\).
  4. Review the output: inspect the fitted equation, coefficients \(a\) and \(b\), correlation, \(R^2\), standard error and the residuals table. The summary also shows sample size and basic data ranges.
  5. Make predictions: enter a positive value of \(x\) in the prediction box to obtain a fitted value \(\hat y(x)\). If \(x \le 0\), the calculator shows an informative message instead.
  6. Interpret results: use \(R^2\) to judge how well the model fits the data and examine the residuals to spot patterns that might indicate a poor model choice.

6. When Is Logarithmic Regression Appropriate?

Logarithmic regression is particularly useful when the relationship between the response and the predictor is monotonic but with a decreasing marginal effect. Common examples include:

  • Diminishing returns: situations where increasing an input still increases the output, but each extra unit of input has a smaller effect than the previous one.
  • Scale effects: models where a change from \(x = 1\) to \(x = 2\) has a much larger impact than a change from \(x = 101\) to \(x = 102\), even though both changes are +1.
  • Transform-to-linear strategies: using \(\ln(x)\) to linearize a nonlinear relationship so that standard regression tools apply.

If the pattern in your scatter plot of \(y\) versus \(x\) curves downward and seems roughly straight when you plot \(y\) against \(\ln(x)\), a logarithmic regression model can be a good choice. In other situations, a linear, exponential, power or polynomial model may be more appropriate.

7. Assumptions and Limitations

Like other regression techniques, logarithmic regression relies on several assumptions:

  • Positive predictor values: all \(x_i\) must be strictly positive so that \(\ln(x_i)\) is defined.
  • Linear relationship in the transformed scale: the model assumes a roughly linear relationship between \(y\) and \(z = \ln(x)\).
  • Independent errors: residuals should be approximately independent across observations.
  • Constant variance: the variability of the residuals should not change dramatically across the range of \(\ln(x)\).

The calculator is designed for exploratory data analysis, educational use and many applied settings. For high-stakes decisions, it is a good idea to combine numerical output with diagnostic plots and subject-matter expertise.

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Logarithmic Regression Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about the logarithmic model \(y = a + b \ln(x)\), when to use it and how to interpret the calculator output.

Use logarithmic regression when the relationship between \(y\) and \(x\) is not straight but becomes more linear after transforming \(x\) with a natural logarithm. If a scatter plot of \(y\) versus \(x\) curves downward but a plot of \(y\) versus \(\ln(x)\) looks roughly linear, the model \(y = a + b \ln(x)\) is often appropriate. If the original data already show a near-straight-line pattern, a standard linear regression might be sufficient.

The natural logarithm \(\ln(x)\) is only defined for \(x > 0\). Because the model and the regression formulas use \(z = \ln(x)\) as the predictor, any zero or negative value of \(x\) would make the calculations invalid. The calculator therefore checks that all \(x\)-values are strictly positive and alerts you if this requirement is not met.

In a logarithmic regression, \(R^2\) has the same interpretation as in linear regression: it measures the proportion of the variability in the response \(y\) that is explained by the model, after transforming the predictor to \(\ln(x)\). Values of \(R^2\) closer to 1 indicate that the fitted curve tracks the data more closely, while values near 0 indicate that the model explains little of the variation in \(y\).

The calculator uses double-precision arithmetic, so it can handle a wide range of positive \(x\)-values. However, extremely small or extremely large values can lead to numerical instability or make the model difficult to interpret. It is generally good practice to work in a reasonable scale, and, if necessary, rescale your data before fitting the model and then adjust the interpretation accordingly.

Yes. You can enter the same \((x, y)\) data used in your homework or project and compare the calculator’s coefficients, equation and \(R^2\) with your own calculations. This is especially helpful for verifying numerical work. For written solutions, you should still show the underlying formulas and steps, since the calculator provides the final numbers rather than symbolic derivations.

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