Exponential Growth and Doubling Notes

Doubling represents a fundamental pattern of exponential growth.
General continuous exponential growth equation: $A(x) = P e^{r x}$
Mathematical representation of doubling per unit step (where $x$ represents the number of folds or doubling iterations): $A(x) = 2^x$

Handwritten lecture notes detailing exponential growth equations, logarithmic solution for paper folding to Mars, and compounding formulas

Thickness of 1 standard sheet of paper: $\approx 0.1\,\text{mm}$
Average distance between Earth and Mars: $225,000,000\,\text{km} = 2.25 \times 10^{11}\,\text{m}$
Question: How many paper folds ( $x$ ) are required for the thickness of a folded sheet of paper to equal the distance from Earth to Mars?
Equation setup: $2.25 \times 10^{11} \approx 2^x$
Taking the natural logarithm of both sides: $\ln(2.25 \times 10^{11}) = \ln(2^x)$
Applying the power rule for logarithms: $\ln(2.25 \times 10^{11}) = x \ln(2)$
Solving explicitly for $x$ : $x = \frac{\ln(2.25 \times 10^{11})}{\ln(2)}$
Numerical calculation: $x = 50.99... \approx 51\,\text{folds}$

Span of the observable universe: $93\,\text{billion light years}$
Number of paper folds required to reach the scale of the observable universe: $\approx 90\,\text{folds}$

Annual growth base model: $P e^{r t}\quad (\text{annually})$
Continuous exponential growth model: $P e^{r t}\quad (\text{continuous})$
Discrete periodic compound growth model: $P\left(1 + \frac{r}{n}\right)^{nt}\quad (\text{discrete})$