1. Notes 6.6, DATE___________
The Natural Base, e

2. Natural Exponential Functions
Model situations in which a quantity grows or decays continuously
investments

3. What is “e”? 2.718281828… Find “e” on your calculator Should be 2nd ÷
Press enter… what do you have? …

4. “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

5. “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=

6. Examples cont…
Ex) 3e = x
x= 3.154
Ex) 3e = x
x=2.320

7. Continuous Interest Formula: A = Pert Interest paid at every moment
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 = $

8. 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

9. Examples Ex) x = 3 y = ln 3 y = 1.099 Ex) x = 1/8 y = ln 1/8 y = -2.08
Evaluate y = ln x
Ex) x = 3
y = ln 3
y = 1.099
Ex) x = 1/8
y = ln 1/8
y = -2.08

10. Natural Log Properties
1) eln x = x
2) ln ex = x
Ex) eln3 = 3
Ex) e4ln4 =
= eln4
= 44
= 256 4
Ex) ln e7 = 7
Ex) 5 ln e2 =
= 5 * 2
= 10

11. Write an equivalent exponential or logarithmic expression
ex = y  x = ln y
1) ex = 30
x = ln 30
2) e0.69 ≈ 1.99
ln 1.99 ≈ 0.69

12. Solving using natural logs
Similar to solving with logs…take the ln of both sides to solve
1) 35x = 30
2) 13x = 27
ln 35x = ln 30
ln 13x = ln 27
x ln 35 = ln 30
x ln 13 = ln 27
x = ln 30
ln 35
x = ln 27
ln 13
x =
x = 1.29