Exponential functions with base e

A General Note: The Number e

The letter e represents the irrational number

[latex]{\left(1+\frac{1}{n}\right)}^{n},\text{as}n\text{increases without bound}[/latex]

The letter e is used as a base for many real-world exponential models. To work with base e, we use the approximation, [latex]e\approx 2.718282[/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.

Example: Using a Calculator to Find Powers of e

Calculate [latex]{e}^{3.14}[/latex]. Round to five decimal places.

Try It

Use a calculator to find [latex]{e}^{-0.5}[/latex]. Round to five decimal places.

Investigating Continuous Growth

So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, e is used as the base for exponential functions. Exponential models that use e as the base are called continuous growth or decay models. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.

A General Note: The Continuous Growth/Decay Formula

For all real numbers t, and all positive numbers a and r, continuous growth or decay is represented by the formula

[latex]A\left(t\right)=a{e}^{rt}[/latex]

where

  • a is the initial value,
  • r is the continuous growth rate per unit time,
  • and t is the elapsed time.

If r > 0, then the formula represents continuous growth. If r < 0, then the formula represents continuous decay.

For business applications, the continuous growth formula is called the continuous compounding formula and takes the form

[latex]A\left(t\right)=P{e}^{rt}[/latex]

where

  • P is the principal or the initial invested,
  • r is the growth or interest rate per unit time,
  • and t is the period or term of the investment.

How To: Given the initial value, rate of growth or decay, and time t, solve a continuous growth or decay function.

  1. Use the information in the problem to determine a, the initial value of the function.
  2. Use the information in the problem to determine the growth rate r.
    1. If the problem refers to continuous growth, then r > 0.
    2. If the problem refers to continuous decay, then r < 0.
  3. Use the information in the problem to determine the time t.
  4. Substitute the given information into the continuous growth formula and solve for A(t).

Example: Calculating Continuous Growth

A person invested $1,000 in an account earning a nominal 10% per year compounded continuously. How much was in the account at the end of one year?

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A person invests $100,000 at a nominal 12% interest per year compounded continuously. What will be the value of the investment in 30 years?

Example

Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?

Try It

Using the data in Example 9, how much radon-222 will remain after one year?