The Natural Base e

Section 6.2 beginning on page 304

The Natural Base eThe Natural Base e, or the Euler number, is a number like or .

e is an irrational number and it is defined as follows:

As x approaches , approaches

Example 1: Simplify each expression.

a) b) c)

¿𝑒3+6

¿𝑒9¿ 4𝑒5−4

¿ 4𝑒¿32𝑒2 (− 4 𝑥)

¿9𝑒− 8𝑥

¿9

𝑒8𝑥

Graphing Natural Base FunctionsPg. 305

Graphing Natural Base FunctionsExample 2: Tell whether each function represents exponential growth or exponential decay, then graph the function.

a) ** Because a = 3 (it is positive) and r = 1 (also positive), this is an exponential growth function.

** Use a table of values to graph the function.

X -2 -1 0 1

Y 0.41 1 .10 3 8.15

Graphing Natural Base FunctionsExample 2: b)

** Because a = 1 (it is positive) and r = -0.5 (negative), this is an exponential decay function.

** Use a table of values to graph the function.

X -4 -2 0 2

Y 7.39 2.72 1 0.37

Why did I choose different x values t his time?

Solving Real Life Problems

𝐴=4500𝑒0.04 𝑡

Your friends account has a balance of about $7,300 after 10 years.

𝐴=4500𝑒0.04 (10 )

𝐴=4500𝑒4

𝐴≈$6,713.21

Since when t=0 the balance is 4000, your principal is greater than your friends because you started with 4500.

Your friend has a greater balance after 10 years.

Monitoring ProgressSimplify the expressions:

1) 2) 3)

Tell whether the function represents exponential growth or decay then graph the function.

4) 5) 6)

¿𝑒11 ¿3𝑒3 ¿1000

𝑒9𝑥

Monitoring Progress7) You deposit $4250 in an account that earns 5% annual interest compounded continuously. Compare the balance after 10 years with the accounts in example 3.

Friends account in example 3: Balance after 10 years $7,00Your account in example 3: Balance after 10 years

This account will have a balance of $7007.07, which is greater than your original account, but still less than your friends account.