Logarithms 7-6 The natural base, e.

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Logarithms 7-6 The natural base, e

7-6 The natural base, e What do we know about π e is a number like π Irrational number Never repeats: 3.141592654… Very important to geometry and circles e is a number like π Irrational number Never repeats: 2.718281284590… Very important to business and finance

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

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

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 e8 2. ln ex 3. ln ex+2 4. ln e3x

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 5 6. eln 2x 7. eln x-7 8. eln 0.5

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 9. e3ln 5 10. e7ln x 11. e2ln (x+2) 12. e0.5ln x

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?

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

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 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.

HW 7-6 Practice B worksheet