Exponential and Logarithmic Modeling

Many real-world phenomena grow or decay at rates proportional to their current size, or span such massive ranges that standard linear scales fail. We use exponential and logarithmic functions to model these situations accurately.

Exponential Growth and Decay

The standard model for continuous exponential growth or decay is:

$A(t) = A_0 e^{kt}$

• $A(t)$ is the amount after time $t$.
• $A_0$ is the initial amount (when $t = 0$).
• $k$ is the continuous rate of growth ($k > 0$) or decay ($k < 0$).
• $t$ is time.

Example: Continuous Compounding

Problem: An investment of $5000 grows at 4% compounded continuously. Find the value after 10 years.

Solution: Use the continuous compounding formula $A = P e^{rt}$:

• $P = 5000$
• $r = 0.04$
• $t = 10$

$A = 5000 e^{0.04 \times 10}$ $A = 5000 e^{0.4} \approx 5000 \times 1.49182 = 7459.12$

The investment will be worth roughly $7,459.12.

Half-Life Modeling

Half-life is the time required for a decaying substance to reduce to exactly half of its initial amount. The formula for half-life is:

$A(t) = A_0 \left(\frac{1}{2}\right)^{\frac{t}{h}}$

Where $h$ is the half-life duration.

Example: Radioactive Decay

Problem: A substance has a half-life of 8 hours. How long until only 10% remains?

Solution: We want the final amount $A(t)$ to be $0.10 A_0$.

$0.10 A_0 = A_0 \left(\frac{1}{2}\right)^{\frac{t}{8}}$

Divide both sides by $A_0$:

$0.10 = 0.5^{\frac{t}{8}}$

Take the natural logarithm ($\ln$) of both sides to solve for $t$:

$\ln(0.10) = \ln\left(0.5^{\frac{t}{8}}\right)$ $\ln(0.10) = \frac{t}{8} \ln(0.5)$ $t = \frac{8 \ln(0.10)}{\ln(0.5)} \approx \frac{8(-2.3026)}{-0.6931} \approx 26.58$

It will take approximately 26.58 hours for only 10% of the substance to remain.

Logarithmic Modeling

Logarithmic functions are the inverses of exponential functions. They are primarily used when working with values that vary over enormous magnitudes.

• pH Scale: Measures the acidity of a solution based on hydrogen ion concentration: $\text{pH} = -\log[H^+]$
• Richter Scale: Measures the magnitude of earthquakes: $M = \log\left(\frac{I}{I_0}\right)$
• Decibels: Measures sound intensity.

In all these models, an increase of 1 on the logarithmic scale corresponds to a 10-fold increase in the actual physical quantity (since they typically use base-10 logarithms).