1. Base e and Natural Logarithms
Evaluate using the following table n 10
100
1000
10000
100000 e

2. Base e and Natural Logarithms
10—2.5937
100—2.7048
1000—2.7169
10000—2.7181
100000—2.7183
—2.7183
e= …

3. Base e and Natural Logarithms
Natural Base e
ln x = loge x
The system of natural logarithms has the number called e as its base. (e is named after the 18th century Swiss mathematician, Leonhard Euler.)  e is the base used in calculus.  It is called the "natural" base because of certain technical considerations.
It is used in continual growth/decay situations

4. Base e and Natural Logarithms
Inverse Property of Base e and Natural Logarithms e ln x = x and ln ex = x
All rules for log’s apply for ln’s
ln 732=
2.5958
ln =

5. Base e and Natural Logarithms
A=A0ekt
A = end value, Ao=beginning value
After 500 years, a sample of radium-226 has decayed to 80.4% of its original mass. Find the half-life of radium-226.
0.804 = 1e500k
ln ( 0.804) = ln (e500k)

6. ln ( 0.804) = 500k ln e ln e=1
ln ( 0.804) = 500k
k= (ln 0.804)/500.
This is the exact solution; evaluate the natural log with a calculator to get the decimal approximation k =

7. Base e and Natural Logarithms
Since we now know k, we can write the formula (function) for the amount of radium present at time t as
A=A0 e t

8. Base e and Natural Logarithms
Now, we can finally find the half-life. We set A=1/2 A0 and solve for t.
(1/2)A0=A0 e t
1/2 = e t
ln(1/2) = ln[e t].
ln (1/2) = t
t= ln(1/2) /( )
t = 1590

9. Base e and Natural Logarithms
2x=64 solve using ln
X ln 2= ln 64
X=ln 64/ln 2
X=6

10. Base e and Natural Logarithms
POPULATION The equation A = A0e rt describes the growth of the world’s population where A is the population at time t, A0 is the population at t =0, and r is the annual growth rate. How long will take a population of 6.5 billion to increase to 9 billion if the annual growth rate is 2%?
9=6.5e.02t
t=16.3 years

11. Base e and Natural Logarithms
INTEREST Horatio opens a bank account that pays 2.3% annual interest compounded continuously. He makes an initial deposit of 10,000. What will be the balance of the account in 10 years? Assume that he makes no additional deposits and no withdrawals.
Use A = A0e rt
A=10000e .023(10)
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