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.