7.3 The Natural Base e

The Natural Base e

A famous constant (similar to π) is the natural base, e (also known as Euler’s number).

e ≈ 2.718281828459…e is irrational (non-repeating, not-terminating).A function in the form f(x) = aebx is a natural base

exponential function.if b>0…growthif b<0…decay

Let’s take a look at the graph of a natural base exponential function…

Ex. Graph f(x) = ex

x f(x)

–2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Ex. Graph f(x) = ex

x f(x)

–2 -1.5 -1 -0.5 0 0.5 1 1.5 2.14 .22 .37 .61 1 1.65 2.72 4.48 7.39

Simplify.a. e3 • e5

b. 8e6 2e2

c. (2e–3x)–2

= e8

= 4e4

= 2–2e6x

= e6x

4

To simplify natural base expressions, follow the Properties of Exponents:

Recall that we have studied the formula for the value of an investment in which interest is compounded n times per year:

A = P(1 + r/n)nt

Here is another common investment formula:A = Pert

A is the balanceP is the initial principalr is the interest ratet is the number of years

In this formula, the interest is compounded continuously (about every second)

Example: For a deposit of $5000 into an account that pays 8.5% interest, find the balance after 5 years if the interest is compounded continuously.

A = Pert

= 5000 • e.085 • 5 = $7647.95

Homework:

p.495-497 #3-39 multiples of 3, #56 and #57