The Natural The Natural Base, e Base, e

Notes 6.6, DATE___________

Natural Exponential Functions

• Model situations in which a quantity grows or decays continuously– investments

What is “e”?

•Find “e” on your calculator

•Should be 2nd ÷

•Press enter… what do you have?

2.718281828…

“e”

• e is an irrational number like pi

– It keeps going on FOREVER

– Instead of rounding or writing out a ton of numbers, just use your calculator

“e” with an exponent

• Somewhere on your calculator, you should have the button ex

• Should be 2nd LN on your graphing calculator • Just press 2nd LN and insert the exponent then

press enter (round to the nearest thousandth)

Example: e6 = x

x= 403.429

Examples cont…

Ex) 3e0.05 = x

Ex) 3e-0.257 = x

x=2.320

x= 3.154

Continuous Interest• Interest paid at every moment

Formula: A = Pert

Ex) What is the amount of an investment starting with $1000 invested at 3% for ten years where interest is paid continuously.

A = Pert

A = (1000)e(.03)(10)

A = $1349.86

Natural Logs y = lnx

• A Natural Log is an INVERSE OPERATION of “e”

• You can find Natural Log labeled as LN on your calculator

• Just like when we worked on e, just press LN and plug in your numbers

Examples

Ex) x = 3

y = ln 3 y = 1.099

•Ex) x = 1/8

y = ln 1/8y = -2.08

Evaluate y = ln x

Natural Log Properties2) ln ex = x

Ex) eln3 =

1) eln x = x

Ex) ln e7 =

Ex) e4ln4 = = eln4

= 44

= 256

Ex) 5 ln e2 = = 5 * 2 = 10

4

3

7

Write an equivalent exponential or logarithmic expression

1) ex = 30x = ln 30

2) e0.69 ≈ 1.99

ln 1.99 ≈ 0.69

ex = y x = ln y

Solving using natural logs

1) 35x = 30 2) 13x = 27

x = 0.9566

Similar to solving with logs…take the ln of both sides to solve

ln 35x = ln 30x ln 35 = ln 30

x = ln 30 ln 35

ln 13x = ln 27x ln 13 = ln 27

x = ln 27 ln 13

x = 1.29