How do you do exponential regression by hand?

Take ln of each y, run linear regression of ln y on x to get intercept c and slope k, then a = eᶜ and b = eᵏ, giving y = a·bˣ.

Why must y values be positive?

The fit uses ln y, which is undefined for zero or negative numbers. Shift the data or use another model if some y values are not positive.

Is this the same as TI-84 ExpReg?

Yes. ExpReg fits y = a·bˣ by linear regression on ln y, and the a, b and r values match.

How do I find the growth rate?

Growth rate per unit of x = (b − 1) × 100%. With b = 1.35, the quantity grows 35% per step.

Exponential Regression Calculator

Home › Statistics & Probability Calculators › Exponential Regression Calculator

Exponential Regression Calculator

This exponential regression calculator fits the model y = a·bˣ to your data. It takes the natural log of every y value, fits a straight line to (x, ln y), and converts the intercept and slope back to a and b. The same result is shown in the continuous form y = a·e^kx with k = ln b.

Exponential models describe quantities that grow or shrink by a fixed percentage per step: compound interest, population growth, bacterial counts, radioactive decay and cooling. If b is greater than 1 the quantity grows by (b − 1) × 100% per unit of x and the doubling time is ln 2 ÷ k; if b is below 1 it decays and the half-life is shown instead. All y values must be positive because the method uses logarithms.

The fit matches the TI-84 ExpReg command and Excel’s LOGEST, both of which minimise squared error on the log scale, and r² is reported on that scale as they do. A plot of ln y against x that looks straight confirms the model; you can draw one with the scatter plot maker. For curves with a single bend, compare with the quadratic regression calculator.

x values

Separate values with commas, spaces or new lines.

y values (all positive)

Separate values with commas, spaces or new lines.

Predict y at x = (optional)

Result and step-by-step solution

Exponential model

y = 119.830503 × 1.33899^x

Equivalent: y = 119.830503 e^0.291915x. r = 0.99987, r² = 0.999739 (on ln y, as on the TI-84).

Linearise
Take natural logs: ln y = ln a + x ln b, a straight line in x.
Transformed data
x y ln y fitted y
0 120 4.78749 119.8305
1 162 5.0876 160.4518
2 211 5.35186 214.8433
3 290 5.66988 287.6729
4 382 5.94542 385.191
5 520 6.25383 515.7667
6 690 6.53669 690.6062
Fit the line to (x, ln y)
Intercept ln a = 4.786078, slope ln b = k = 0.291915
Back-transform
a = e^4.786078 = 119.830503; b = e^0.291915 = 1.33899
Prediction
At x = 8: y = 119.830503 × 1.33899⁸ = 1238.183

x	y	ln y	fitted y
0	120	4.78749	119.8305
1	162	5.0876	160.4518
2	211	5.35186	214.8433
3	290	5.66988	287.6729
4	382	5.94542	385.191
5	520	6.25383	515.7667
6	690	6.53669	690.6062

Growth / decay rate per unit of x	33.899%
Doubling time	2.3745 units of x

Frequently asked questions

How do you do exponential regression by hand?: Take ln of each y, run linear regression of ln y on x to get intercept c and slope k, then a = eᶜ and b = eᵏ, giving y = a·bˣ.
Why must y values be positive?: The fit uses ln y, which is undefined for zero or negative numbers. Shift the data or use another model if some y values are not positive.
Is this the same as TI-84 ExpReg?: Yes. ExpReg fits y = a·bˣ by linear regression on ln y, and the a, b and r values match.
How do I find the growth rate?: Growth rate per unit of x = (b − 1) × 100%. With b = 1.35, the quantity grows 35% per step.

Last reviewed: October 7, 2026. Calculations run in your browser and were validated against SciPy.

Exponential Regression Calculator

Result and step-by-step solution

Frequently asked questions

Related calculators