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Exponential & Logarithmic Modeling Lab

Investigate exponential growth, decay, and logarithmic relationships with interactive graphs. Fit models to real data, compare R² values, and discover when each function type is the best choice.

Guided Experiment: Growth vs Decay Investigation

How does changing the base b from greater than 1 to between 0 and 1 affect the shape of y = a·b^x? What happens to half-life and doubling time?

Write your hypothesis in the Lab Report panel, then click Next.

Exponential Function y = a·b^x

Adjust a and b to explore growth and decay. The horizontal asymptote y = 0 is shown in red.

Controls

a (initial value)1.0
b (growth/decay factor)1.05
Growth (b > 1)
x min0
x max50

Results

y=1.00⋅1.0500xy = 1.00 \cdot 1.0500^{x}
Continuous Form
y=1.00⋅e0.0488xy = 1.00 \cdot e^{0.0488x}
Type
Growth
k = ln(b)
0.0488
Doubling Time
t₂ = 14.21
y(0)
1.00
y(50)
11.47
t1/2=ln⁡2∣ln⁡b∣t_{1/2} = \frac{\ln 2}{|\ln b|} (decay)  |  t2=ln⁡2ln⁡bt_2 = \frac{\ln 2}{\ln b} (growth)

Data Table

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#TrialModelab / kR²t½ / t₂Residual Sum
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Reference Guide

Exponential Growth & Decay

An exponential function models quantities that change by a constant percentage each period.

y=a⋅bxy = a \cdot b^x

When b > 1 the function grows without bound. When 0 < b < 1 it decays toward zero. The initial value is a (the y-intercept at x = 0).

Half-Life & Doubling Time

These time constants describe how fast an exponential process unfolds.

t1/2=ln⁡2∣ln⁡b∣t2=ln⁡2ln⁡bt_{1/2} = \frac{\ln 2}{|\ln b|} \qquad t_2 = \frac{\ln 2}{\ln b}

Half-life is the time for a decaying quantity to reach half its value. Doubling time is the time for a growing quantity to double. Both depend only on b, not on the starting value a.

Logarithmic Functions

The natural logarithm is the inverse of the exponential function.

y=a⋅ln⁡(x)+cy = a \cdot \ln(x) + c

The domain is x > 0 with a vertical asymptote at x = 0. Logarithmic models fit data that increases rapidly then levels off (decibel scales, learning curves, diminishing returns).

log⁡b(x)=ln⁡(x)ln⁡(b)\log_b(x) = \frac{\ln(x)}{\ln(b)}

Linearization & Curve Fitting

Linearization transforms a nonlinear relationship into a linear one for regression analysis.

ln⁡(y)=ln⁡(a)+bx⇒plot ln⁡(y) vs x\ln(y) = \ln(a) + bx \quad \Rightarrow \quad \text{plot } \ln(y) \text{ vs } x

For exponential fits, plot ln(y) vs x. For logarithmic fits, plot y vs ln(x). The R² value (coefficient of determination) measures how well the model explains the data. Values closer to 1 indicate a better fit.