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7-6 The natural base, e
• Irrational number• Never repeats: 3.141592654…• Very important to geometry and circles
• Irrational number• Never repeats: 2.718281284590…• Very important to business and finance
What do we know about π
e is a number like π
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: ex
Graph of: 2x
Graph of: 3x
7-6 The natural base, e
Example 1: Graph f(x) = ex-2 + 1
x f(x)
-3 1.0
-2 1.0
-1 1.0
0 1.1
1 1.3
2 2
3 3.7
7-6 The natural base, eA 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 e8 2. ln ex 3. ln ex+2 4. ln e3x
7-6 The natural base, eA 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 5 6. eln 2x 7. eln x-7 8. eln 0.5
7-6 The natural base, eA 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 5 10. e7ln x 11. e2ln (x+2) 12. e0.5ln x
7-6 The natural base, eWord problems involving Economic Applications!
Growth decay formula: A(t) = a(1±r)t
Compound interest formula: A= Pert
A = total amount
r = rate (percent)
P = Principal (initial amount)
t = time
Example 13. What is the total amount for an investment of $500 invested at 5% for 40 years and continuously compounded?
7-6 The natural base, eWord problems half-life!
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
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
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
Part B: Find the number of years that answers the original question.
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?