Notes on Exponential Modeling & Solving using Logs

Exponential Modeling Formula

  • The exponential model is represented by the formula: y = a \cdot b^x

    • y represents the dependent variable.
    • a represents the initial amount or y-intercept.
    • b represents the common ratio.
    • x represents the independent variable, often time.

Letters Meaning

  • a: y-intercept; starting amount.
  • b: 1 ± %; common ratio.

Use When

  • y = a \cdot b^x can be used when time is in units other than years, such as hours.

Compound Interest Formula

  • The formula for compound interest is: A(t) = P(1 \pm r)^t

    • A(t) represents the future amount.
    • P represents the principal or starting amount.
    • r represents the interest rate, converted to decimal form.
    • t represents the time in years.

  • This formula is used for money that is compounding annually (once per year).

Compounding more than Yearly

  • The formula for compounding more than yearly but less than continuously is: A(t) = P(1 + \frac{r}{n})^{nt}

    • A(t) represents the future amount.
    • P represents the principal or starting amount.
    • r represents the interest rate, converted to decimal form.
    • n represents the frequency in which money compounds (e.g., quarterly n = 4, monthly n = 12, weekly n = 52, daily n = 365).
    • t represents the time in years.

Compounding Continuously

  • The formula for compounding continuously is: A(t) = Pe^{rt}

    • A(t) represents the future amount.
    • P represents the principal or starting amount.
    • e is the base of the natural logarithm (approximately 2.71828).
    • r represents the interest rate, converted to decimal form.
    • t represents the time in years.