Logarithmic Regression TI-84: How to Calculate

This page explains how to perform logarithmic regression on a TI-84 graphing calculator. You enter your x and y data into lists, run the logarithmic regression command, and get the equation y = a + b·ln(x) along with the correlation coefficient r². The steps below apply to the TI-84 Plus, TI-84 Plus CE, and similar models.

Formula

The logarithmic regression model is:

y = a + b·ln(x)

• y – the dependent variable (response)
• x – the independent variable (predictor), must be positive
• a – the intercept (value of y when ln(x) = 0, i.e., x = 1)
• b – the slope (change in y per unit change in ln(x))
• ln – natural logarithm (base e)

The TI-84 computes a and b by the least-squares method after transforming each x to ln(x). It reports r² (coefficient of determination) and r (correlation coefficient) if diagnostics are on.

Worked Example

Suppose you have data (1, 2), (2, 3), (3, 4), (4, 5). Using the default values in the calculator above, the regression yields:

1. Transform x: ln(1)=0, ln(2)=0.693, ln(3)=1.099, ln(4)=1.386
2. Fit a straight line to (ln(x), y): the least-squares line is y = 2 + 1·ln(x)
3. So a = 2, b = 1, and r² = 1 (perfect fit for this artificial data)

The calculator above shows the same result: a = 2, b = 1, r² = 1.0000.

When to Use Logarithmic Regression

Logarithmic regression fits data where y increases (or decreases) rapidly at first, then levels off. Common examples:

• Learning curves: performance improves quickly then plateaus
• Sound intensity vs. perceived loudness (decibels)
• Population growth with limited resources (logistic is better, but log is a simple approximation)
• Dose-response relationships in biology

If your scatterplot shows a curve that rises steeply then flattens, try logarithmic regression. If it rises exponentially, use exponential regression instead.

Step-by-Step: TI-84 Procedure

1. Press STAT and choose Edit to enter data into lists L1 (x) and L2 (y).
2. Press 2nd then QUIT to return to the home screen.
3. Press STAT, move to the CALC menu.
4. Scroll to LnReg (usually option 9 on the TI-84 Plus).
5. Press ENTER. The command LnReg appears.
6. Enter L1, L2 – press 2nd 1, comma, 2nd 2.
7. Optionally, add , Y1 to store the equation into Y1 for graphing.
8. Press ENTER to execute. The calculator displays a, b, r², and r.

If r² or r do not appear, turn on diagnostics: press 2nd CATALOG, scroll to DiagnosticOn, press ENTER twice, then run LnReg again.

Common Problems and Fixes

• Error: Domain – This occurs if any x value is zero or negative. Logarithmic regression requires x > 0. Remove or transform such values.
• r² not showing – Diagnostics are off. Turn on DiagnosticOn as described above.
• Equation not graphing – You didn’t store the equation in Y1. Re-run LnReg with L1, L2, Y1.
• Poor fit (low r²) – The data might not follow a logarithmic pattern. Try other models (quadratic, power, exponential).
• List length mismatch – L1 and L2 must have the same number of entries. Check for missing values.

Comparing Logarithmic and Other Regressions

Logarithmic regression is one of several curve fits available on the TI-84. Here’s how it compares:

• Linear (LinReg): y = a + bx. Use when the scatterplot looks like a straight line.
• Exponential (ExpReg): y = ab^x. Use when y grows or decays by a constant percentage.
• Power (PwrReg): y = ax^b. Use when both variables are transformed logarithmically.
• Quadratic (QuadReg): y = ax² + bx + c. Use when the curve has one bend (parabola).
• Logistic (Logistic): y = c/(1+ae^(-bx)). Use for S-shaped growth with an upper limit.

To decide, plot the data and see which pattern matches. You can also compare r² values – higher is generally better, but beware of overfitting with more complex models.

Limitations and Accuracy

Logarithmic regression assumes a specific functional form. It cannot model data that decreases then increases, or data with multiple inflection points. Also, the model is undefined for x ≤ 0, so such data must be excluded or shifted. The TI-84 uses least-squares estimation, which is sensitive to outliers. Always plot the residuals or at least the fitted curve to visually verify the fit.

Rounding: the calculator stores full precision internally, but displays 4 decimal places by default. For more digits, change the display mode (MODE, then Float to 6 or 9). When reporting results, use the unrounded values to avoid errors in predictions.

Frequently Asked Questions

How do I turn on r² on my TI-84?

Press 2nd CATALOG, scroll down to DiagnosticOn, press ENTER twice. Then re-run the LnReg command. The r² and r values will appear.

Why do I get a Domain error when doing LnReg?

The natural logarithm is only defined for positive numbers. Check your x-list for zeros or negative values. Remove those points or add a constant to shift all x values to be positive.

Can I use logarithmic regression if my x-values are negative?

No, not directly. You would need to transform the data first, for example by adding a constant to make all x > 0. The model would then be y = a + b·ln(x + c), which is not the standard LnReg on the TI-84.

What does the r² value mean?

r² (coefficient of determination) tells you how much of the variation in y is explained by the model. An r² of 0.95 means 95% of the variability is captured. Values close to 1 indicate a good fit, but a high r² does not guarantee the model is appropriate – always check the scatterplot.

How do I predict y for a new x using the regression equation?

Once you have a and b, plug the new x into y = a + b·ln(x). On the TI-84, store the equation in Y1, then use 2nd TRACE (CALC) and choose value, or just type the expression on the home screen.

Is logarithmic regression the same as natural log regression?

Yes, on the TI-84, LnReg uses the natural logarithm (base e). If you need base-10 log, you would transform x manually (log10(x) = ln(x)/ln(10)) and then do a linear regression on the transformed data.

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in x | x value | - | 1 | 0
in y | y value | - | 2 | 0
out a | Intercept a | - | 2 | 2
out b | Slope b | - | 1 | 2
out r2 | R-squared | - | 1 | 4