Logarithms7-6 The natural base, e

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?

HW 7-6 Practice B worksheet