1. Logarithms
7-6 The natural base, e

2. 7-6 The natural base, e What do we know about π e is a number like π
Irrational number
Never repeats: …
Very important to geometry and circles
e is a number like π
Irrational number
Never repeats: …
Very important to business and finance

3. 7-6 The natural base, e e is a number: 2.718281284590…
because it is a number we can draw a graph of it.
Graph of: 2x
Graph of: ex
Graph of: 3x

4. 7-6 The natural base, e Example 1: Graph f(x) = ex-2 + 1 x f(x) -3 1.0-1
1.1 1
1.3 2 3
3.7

5. 7-6 The natural base, e
A log with a base of e is called a natural logarithm (ln)
All the things we can do with logs we can do with natural logs
1. ln e ln ex ln ex ln e3x

6. 7-6 The natural base, e
A log with a base of e is called a natural logarithm (ln)
All the things we can do with logs we can do with natural logs
5. eln eln 2x eln x eln 0.5

7. 7-6 The natural base, e
A log with a base of e is called a natural logarithm (ln)
All the things we can do with logs we can do with natural logs
9. e3ln e7ln x 11. e2ln (x+2) e0.5ln x

8. 7-6 The natural base, e Word problems involving Economic Applications!
Growth decay formula: A(t) = a(1±r)t
Compound interest formula: A= Pert
A = total amount
P = Principal (initial amount)
r = rate (percent)
t = time
Example 13. What is the total amount for an investment of $500 invested at 5% for 40 years and continuously compounded?

9. 7-6 The natural base, e Word problems half-life!
Half life formula: N(t) = Noe-kt
N(t) = the amount of material remaining
No = the initial amount of material
k = decay constant
t = time

10. Half life formula: N(t) = Noe-kt
N(t) = the amount of material remaining
k = decay constant
No = the initial amount of material
t = time
Example 14. Plutonium-239 (Pu-239) has a half-life of 24,110 years. How long does it take for a 1 gram sample of Pu-239 to decay to 0.1 gram?
Part A: find the decay constant, k

11. Half life formula: N(t) = Noe-kt
N(t) = the amount of material remaining
k = decay constant
No = the initial amount of material
t = time
Example 14. Plutonium-239 (Pu-239) has a half-life of 24,110 years. How long does it take for a 1 gram sample of Pu-239 to decay to 0.1 gram?
Part B: Find the number of years that answers the original question.

12. HW 7-6 Practice B worksheet