Exponential Growth and Logarithmic Functions Study Notes

  • Starting Point

    • Begin with 200 cells.

    • Time progresses in minutes; amount of cells is the variable of interest.

    • Every 3 minutes, the number of cells doubles.

  • Doubling Function

    • At 3 minutes:

    • Cells = 200 * 2 = 400

    • At 6 minutes:

    • Cells = 200 * 4 = 800

    • General formula for the amount of cells after time t minutes:

    • Formula: 200×2t/3200 \times 2^{t/3}

    • Explanation: Each doubling event occurs every 3 minutes; thus for time t, the exponent is t/3t/3.

  • Alternative Growth Scenarios

    • Growth can be by tripling or quadrupling similarly.

    • Example: \n - Tripling: would use a base of 3 instead of 2 in the formula.

    • Half-Life: Example of radioactive decay.

    • Half-life is time taken for a substance to reduce to half its quantity.

    • Example:

      • Start with 24 mg; after 25 years, it halves:

      • After 25 years: 12 mg

      • After 50 years: 6 mg

      • General formula: 24×12t2524 \times \frac{1}{2}^{\frac{t}{25}}

  • Exponential Growth Overview

    • Special number: Euler’s number (denoted as e).

    • Approx. e = 2.718

    • Appears frequently in calculus, representing natural growth rates.

    • Graphs of exponential functions represented by different bases (e.g., 2, 3, 4) show varying growth rates.

    • The slope of exe^x at y-intercept is equal to e.

  • Logarithmic Functions Overview

    • The log function is the inverse of the exponential function.

    • For example, 103=100010^3 = 1000, therefore log base 10 of 1000 is 3.

    • Basic properties:

    • log(xy) = log(x) + log(y) (sum to product)

    • log(x/y) = log(x) - log(y) (difference to quotient)

    • log(x^k) = k * log(x) (power rule)

    • Natural log (ln) uses base e; it's a commonly used notation.

  • Base Change Formula

    • Allows rewriting of logarithms with different bases.

    • Formula of form: log⁡b(x)=ln⁡(x)ln⁡(b)\log_b(x) = \frac{\ln(x)}{\ln(b)}

    • Useful in simplifying calculations in calculus.

    • Example: Convert 282^8 to natural log:

    • ln⁡(28)=8ln⁡(2)\ln(2^8) = 8 \ln(2)

  • Solving Equations Involving Logarithms

    • Use properties to simplify and isolate logarithms.

    • If given ln⁡(a)=b\ln(a) = b, use eb=ae^{b} = a to eliminate the log.

    • Combining logs and their transformations makes manipulation easier.

    • Steps:

    1. Apply log properties.

    2. Isolate logs.

    3. Exponentiate to solve for variables.

  • Practical Applications

    • Understanding exponential growth is key in various fields such as biology, finance, and physics.

    • The base of natural logarithm e plays a crucial role in calculus, especially in understanding growth rates and decay processe