Welcome to Week 6: The Power of Exponentials

📢 Important Logistics Update: Before we dive into the math, please remember that we have permanently moved classrooms! As of this week, we are meeting in Room D222 (we are no longer in D206).

Today's class was all about connecting the abstract rules of logarithms to the real world. We focused on Sections 4.4 and 4.5, moving from solving equations to creating models that predict future outcomes.

Key Concepts from Class

1. Solving Exponential and Logarithmic Equations (Section 4.4)

In previous weeks, we learned the properties of logs. Now, we use them to solve for variables trapped in the exponent. Remember the Golden Rule of solving these equations: Isolate the exponential term first.

For example, to solve for $x$ in an equation like $5e^{2x} = 20$:

  • Step 1: Divide by 5 to isolate the base. $$ e^{2x} = 4 $$
  • Step 2: Take the natural log ($\\ln$) of both sides (since $\\ln$ is the inverse of $e$). $$ \\ln(e^{2x}) = \\ln(4) $$
  • Step 3: Use the inverse property to bring the power down. $$ 2x = \\ln(4) $$
  • Step 4: Solve for $x$. $$ x = \\frac{\\ln(4)}{2} \\approx 0.693 $$

2. Exponential Modeling (Section 4.5)

This is where the math comes to life. We looked at how to model data involving uninhibited growth and decay. The universal formula we use for continuous growth and decay is:

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

Where:

  • $A(t)$ is the amount at time $t$.
  • $A_0$ is the initial amount (at $t=0$).
  • $k$ is the relative growth rate (if $k > 0$) or decay rate (if $k < 0$).
  • $t$ is time.

We discussed scenarios like bacterial growth (doubling time) and radioactive half-life (carbon dating). The key to these word problems is usually finding the rate $k$ first using given data points, and then using that complete model to answer questions about the future.

Resources & Action Items

  • Class Notes: Download the full PDF for the step-by-step solutions we did on the board: 11-02-25 Chapter 4-4 to 4-5
  • Week 6 Quiz: Don't forget to complete your weekly assessment to test your understanding of these models: Week 6 Quiz Link

Keep practicing those properties of logarithms—they are the key to unlocking these exponential models! See you all in Room D222 next class.