1. C2D8 Bellwork: Fill in the table Fill in the blanks on the worksheet
Recall the formula for compound interest:
P =
n =
r =
t =
Initial value (or amount)
Number of compoundings
Interest rate as a decimal
Time
To develop an equation to determine continuously compounded interest, let P = 1, r = 100% = 1, and t = 1. Let n  n
A(t)
Yearly
n = 1 2
Quarterly
n =
Fill in the table

2. e 2.718281828 Now we will use e in logs!! 2 1 4 2.44141 12 2.61304 365
A(t)
Yearly
n =
Quarterly
Monthly
Daily
Hourly 2 1 4 12
365
8760 e
_________  ______________
Now we will use e in logs!!

3. e
_____  _____________
This number is a constant (like ___), and is referred to as the _________ base. The logarithm with base e is called the _________ logarithm, ___. π
natural
natural ln

4. ln x Sketch y = f(x) for f(x) = ex. f(x) e-1 ≈ 0.37 e0 = 1 ≈ 2.72 e1
y = x x
f(x)
(exact)
(approximate) –1 1 2
f(x) = ex 4 6 8 10 2 –2 –4 –6 –8
–10 y
e-1
≈ 0.37
y = f –1(x) e0
= 1 x
≈ 2.72 e1
≈ 7.39 e2
Sketch y = f–1(x) and y = x.
ln x
f–1(x) =

5. Definition: Whenever you convert back and forth between exponential and logarithmic form using base e, follow the same definition of logs:
Why do base e and natural logs exist?
In the “real” world, you will see examples of both in situations involving growth and decay of natural organisms, population growth, continuous compounding, etc.

6. Never write “e” in the base for the final answer!
Practice #1:
1. Rewrite each equation in logarithmic form.
exponent a) b) c)
argument
log x
= 4
log 5
= 9x
base e e
exponent
log 4
= x e
base
argument
Never write “e” in the base for the final answer!

7. Practice #1:
1. Rewrite each equation in logarithmic form. d) e) f)
log 10
= x – 8
log 16
= 3x + 7
log x5
= 2 e e e

8. 2. Rewrite each equation in exponential form.
a) ln x = 2
b) ln 3 = x
c) ln x2 = 2 2 x 2 e
= x e
= 3 e
= x2
d) ln (x – 4) = 3
e) x = ln 56
f) ln 9 = x4 3 x x4 e
= x – 4 e
= 56 e
= 9
What’s the base?

9. Same thing as before except the base is e!
Properties: Natural logs have the same properties as regular logarithms:
Property:
Logs
Natural Logs
Product
Quotient
Power
Practice #2:
Same thing as before except the base is e!
3. Condense each expression. a) b)

10. 4. Expand the expression. a) b)
5. Simplify each expression.
a) ln e
b) ln e2
c) ln e x + 4
d) eln 5
e) eln (x – 1)
(x + 4) ln e
logee
2 ln e
x – 1 5
e to what power is e?
2 (1)
x + 4
Remember it’s as if the bases “cancel” 2 1

11. Change Of Base: The change of base formula also works for natural logs
Change Of Base: The change of base formula also works for natural logs. Find each to 2 decimal places.
Using base 10
Using base e 30 30 1)
2.45
2.45 4 4 2)
Sometimes there are rounding errors, so ALWAYS do the entire problem in the calculator, then approximate at the end.

12. Rewrite these with exponents
Practice #3: Solve for the variable. Be careful! All types of logarithmic equations are included below, but you will work with natural logs. Check your solutions.
 Definition of Logs:
Rewrite these with exponents
+ 5
+ 5
ln (2x + 3) = 3 3
2x + 3 = e
Exact and approximate
answers are both required!
– 3
– 3
2x = e3 – 3
(exact)
(approx.)
x ≈ 8.54

13.  Property of Equality: ln ( ) = ln ( )
Set each argument equal
Did you check?
ln x2 = ln (2x + 24)
Only x = 6,
Sucka!
x2 = 2x + 24
x2 – 2x – 24 = 0
(x – 6)(x + 4) = 0
x – 6 = 0 or
x + 4 = 0
x = 6
x = –4

14.  Log Properties: Product, Quotient, and Power
Simplify using the properties
Did you check?
Only x = 9
Sucka!

15. Combine, then rewrite.  ln( ) + ln( ) = # Did you check? e x ≈ 36.954 e
(exact)
x ≈ 36.95
(approx.)

16. If you see an “e”, use “ln”
 Logging both sides a)
If you see an “e”, use “ln”
(exact)
(approx.)
x ≈ –2.91
Be careful here…
“ln” before subtracting.
Do not write ln(16).

17. Continuously Compounded Interest Formula:
Value after continuous compounding
A(t) =
A(t) = _________
P =
Principal Value
r =
Compounding rate (as a decimal)
t =
Time
a) An investment of $100 is now valued at $300. The annual interest rate is 8% compounded continuously. About how long has the money been invested?
A(t) =
P =
r =
t =
300
100
0.08
100
100
0.08
0.08
(exact)
What do we do now?
t ≈ years
(approx.)

18. b) An initial deposit of $2000 is now worth $5000
b) An initial deposit of $2000 is now worth $ The account earns 5% annual interest, compounded continuously. Determine how long the money has been in the account.
A(t) =
P =
r =
t =
2000
2000
0.05
0.05
18.33 years