Exponential Regression Calculator

An exponential regression calculator is a powerful mathematical tool used to analyze and model data that exhibits exponential growth or decay.

Imagine you’re studying the growth of a bacterial colony. You measure the number of bacteria at different time intervals:

An exponential regression calculator would use this data to determine the best-fitting exponential function, which might look like: y = 100 * e^(0.46x), where y is the number of bacteria and x is the time in hours.

Exponential Regression Calculation Chart

Sum of (y - y')^2 = 161.87

R-squared = 0.9999 (indicating an excellent fit)

Exponential Regression Formula

The exponential regression formula is typically expressed as:

y = a * e^(bx)

Where:

• y is the dependent variable
• x is the independent variable
• a is the y-intercept (initial value when x = 0)
• b is the growth rate
• e is Euler’s number (approximately 2.71828)

For our bacterial growth example: y = 100 * e^(0.46x)

This formula tells us that the bacterial population starts at 100 and grows exponentially with a rate of 0.46 per hour.

How to Find Exponential Regression?

To find the exponential regression, follow these steps:

1. Collect data: Gather paired (x, y) data points.
2. Linearize the data: Take the natural log of y values to transform the exponential equation into a linear form: ln(y) = ln(a) + bx
3. Perform linear regression: Use the transformed data to find the best-fitting line.
4. Calculate a and b: a = e^(y-intercept), and b = slope
5. Form the exponential equation: y = a * e^(bx)

For example, using bacterial growth data:

Transform y values: ln(100) = 4.61, ln(250) = 5.52, ln(620) = 6.43, ln(1550) = 7.35

Perform linear regression on (x, ln(y)) points

Find y-intercept (4.61) and slope (0.46)

Calculate a = e^4.61 ≈ 100, b = 0.46

Final equation: y = 100 * e^(0.46x)