Transcript What is “e”?

Notes 6.6, DATE___________
The Natural Base, e
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
x= 3.154
Ex) 3e-0.257 = x
x=2.320
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
Evaluate y = ln x
Ex) x = 3
y = ln 3
y = 1.099
•Ex) x = 1/8
y = ln 1/8
y = -2.08
Natural Log Properties
1)
eln x = x
Ex)
eln3
= 3
Ex) ln e7 = 7
Ex) 5 ln e2 =
=5*2
= 10
2) ln ex = x
Ex) e4ln4 =
4
ln4
=e
= 44
= 256
Write an equivalent exponential or logarithmic
expression
x
e
= y  x = ln y
1) ex = 30
x = ln 30
2) e0.69 ≈ 1.99
ln 1.99 ≈ 0.69
Solving using natural logs
Similar to solving with
logs…take the ln of both sides
to solve
1) 35x = 30
ln 35x = ln 30
x ln 35 = ln 30
x = ln 30
ln 35
x = 0.9566
2) 13x = 27
ln 13x = ln 27
x ln 13 = ln 27
x = ln 27
ln 13
x = 1.29