Exponential and Logarithmic Functions

Continuous Compounding Formula

Continuous compounding is a concept in algebra and finance where interest is calculated and added to the principal an infinite number of times per period. The formula for the future value F of an investment with principal P, annual interest rate r, and time t (in years) is:

• Formula:

• Where:

  • F: Future value

  • P: Principal (initial amount)

  • r: Annual interest rate (as a decimal)

  • t: Time in years

  • e: Euler's number, approximately 2.71828

Solving for Interest Rate

To find the interest rate when the future value, principal, and time are known, rearrange the formula:

• Step 1: Start with

• Step 2: Divide both sides by :

• Step 3: Take the natural logarithm (ln) of both sides:

• Step 4: Solve for :

Example Application

Suppose , , and years. Find the interest rate (rounded to the nearest tenth of a percent):

• Substitute into the formula:

• Calculate

• Calculate

• Divide by 10:

• Convert to percent:

Key Terms

• Continuous Compounding: Interest is compounded an infinite number of times per period.

• Natural Logarithm (ln): The logarithm to the base .

• Euler's Number (): A mathematical constant approximately equal to 2.71828.

Summary Table: Continuous Compounding Formula