Exponential Regression TI-84: Steps & Calculator

This page explains how to perform exponential regression on a TI-84 graphing calculator, showing the exact keystrokes and menu paths, and provides a standalone calculator that fits an exponential model to your own (x, y) data. You enter paired data points, and the calculator returns the exponential equation in the form y = a·b^x along with the correlation coefficient r².

Formula

The exponential regression model is y = a · b^x, where:

• y = dependent variable (output)
• x = independent variable (input)
• a = initial value (y-intercept, when x = 0)
• b = base (growth factor if b > 1, decay factor if 0 < b < 1)

The TI-84 fits this by taking the natural log of the y-values, then performing linear regression on (x, ln(y)). The slope of that line equals ln(b), and the intercept equals ln(a). So:

a = e^(intercept), b = e^(slope)

The calculator above uses the same least-squares approach on the logged data. The correlation coefficient r² measures how well the model fits (1.0 is a perfect fit).

Worked Example

Using the default data: x = 1, 2, 3, 4, 5 and y = 2.1, 4.3, 8.9, 18.2, 36.9.

1. Take natural logs of y: ln(2.1)=0.7419, ln(4.3)=1.4586, ln(8.9)=2.1861, ln(18.2)=2.9014, ln(36.9)=3.6082
2. Perform linear regression on (x, ln(y)). The slope is about 0.7145, so b = e^0.7145 ≈ 2.0433.
3. The intercept is about -0.1122, so a = e^(-0.1122) ≈ 0.8939.
4. The model is y = 0.8939 · 2.0433^x.

Checking x=3: 0.8939 · 2.0433^3 ≈ 0.8939 · 8.532 ≈ 7.63, close to the actual 8.9 (the fit is good but not perfect). The calculator block gives a ≈ 0.8939, b ≈ 2.0433, r² ≈ 0.9991.

How to Do Exponential Regression on a TI-84

Here are the exact steps for a TI-84 Plus CE (similar on older TI-84 models):

1. Press STAT, then EDIT, and enter your x-values in L1 and y-values in L2.
2. Press STAT, move right to CALC, then scroll down to ExpReg (option 0 on most models). Press ENTER.
3. The calculator will show ExpReg on the home screen. If you want the equation stored in Y1, press VARS, then Y-VARS, 1: Function, 1: Y1, then press ENTER.
4. Press ENTER again to see the results: a, b, and r² (if diagnostics are on).

If you don’t see r², turn on diagnostics by pressing 2nd then CATALOG (or 0), scroll to DiagnosticOn, press ENTER twice, then re-run the regression.

When to Use Exponential Regression

Use exponential regression when the data shows a constant percentage change per unit increase in x. Common examples include population growth, radioactive decay, compound interest, and bacterial growth. If the data increases by roughly the same factor each step (e.g., doubles every x unit), an exponential model is appropriate. If the data instead increases by a constant amount, linear regression is better. If it grows faster than exponential (e.g., quadratic), try quadratic or cubic regression.

Common Mistakes and Troubleshooting

• Entering data in the wrong lists – double-check that L1 and L2 are selected correctly.
• Using Ln instead of Log – the TI-84’s ExpReg uses natural log (ln), so you don’t need to transform manually.
• Negative or zero y-values – exponential regression requires all y-values to be positive because you can’t take the log of zero or a negative number.
• Not clearing old data – if you have leftover values in L1 or L2, they will affect your regression. Press STAT, EDIT, then highlight the list name and press CLEAR.
• Misreading b – if b is less than 1, it’s decay, not growth. A b of 0.5 means the value halves each step.

Comparing Exponential vs. Other Regressions on TI-84

The TI-84 offers several regression types under STAT > CALC. For exponential data, ExpReg is the direct choice. However, if your data levels off, a logistic model might fit better. If the data has a power-law relationship (y = a·x^b), use PwrReg. The calculator also provides LnReg for logarithmic models. To decide, plot the data and see if it curves upward (exponential), curves downward (logarithmic), or has an S-shape (logistic). The r² value helps compare fits – higher is better, but be wary of overfitting with too many parameters.

Limitations and Accuracy

The exponential regression assumes the relationship is purely exponential. Real-world data often has noise, so r² will be less than 1. The model is only valid within the range of x-values you have; extrapolating far beyond that range can be unreliable. Also, the TI-84 uses a least-squares fit on the logged y-values, which weights smaller y-values more heavily than larger ones. If your data spans several orders of magnitude, consider whether that weighting is appropriate.

Frequently Asked Questions

Why does my TI-84 give an error for ExpReg?

The most common cause is having zero or negative values in your y-list. Exponential regression requires all y-values to be positive. Check your data for zeros, negative numbers, or missing entries (which show as a blank). Also ensure you have at least two data points.

How do I get r² to show on my TI-84?

Press 2nd, then CATALOG (or 0), scroll down to DiagnosticOn, press ENTER twice. After that, every regression will display r and r². If you already have diagnostics on, the values appear automatically.

Can I do exponential regression without a TI-84?

Yes, many online calculators, spreadsheet software like Excel or Google Sheets, and even some scientific calculators can do it. In Excel, you can use the LOGEST function or add a trendline with an exponential fit. The formula is the same: fit a line to (x, ln(y)) and exponentiate.

What’s the difference between exponential and power regression?

Exponential regression models y = a·b^x, where the variable x is in the exponent. Power regression models y = a·x^b, where the variable is the base. To decide, plot your data: if it curves upward and the rate of increase accelerates, try exponential. If it curves but the rate slows down, try power. The TI-84 has both options under STAT > CALC.

Why is my b value negative?

A negative b would imply the function oscillates or flips sign, which is not possible for a simple exponential model with positive y-values. This usually happens when your y-values include negatives or zeros. Fix your data and re-run the regression.

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in x | X values | - | 1, 2, 3, 4, 5 | 0
in y | Y values | - | 2.1, 4.3, 8.9, 18.2, 36.9 | 0
out a | a (initial value) | - | exp((sum(ln(y))*sum(x^2)-sum(x)*sum(x*ln(y)))/(count(x)*sum(x^2)-sum(x)^2)) | 4
out b | b (growth factor) | - | exp((count(x)*sum(x*ln(y))-sum(x)*sum(ln(y)))/(count(x)*sum(x^2)-sum(x)^2)) | 4
out r2 | r² (goodness of fit) | - | (count(x)*sum(x*ln(y))-sum(x)*sum(ln(y)))^2/((count(x)*sum(x^2)-sum(x)^2)*(count(x)*sum(ln(y)^2)-sum(ln(y))^2)) | 4