1. + Natural Logs and More Word Problems

2. + e is a magic number! https://www.youtube.com/watch?v=UFgod5tmLYY

3. + Exponential Functions with e Exponential Functions with a base of e are used to describe CONTINUOUS growth or decay. Some accounts compound interest, every second. We refer to this as continuous compounding.

4. + Continuously Compounded Find the balance in an account paying 3.2% annual interest on $10,000 in 18 years compounded continuously.

5. You put $2000 into an acount earning 4% interest compounded continuously. Find the amount at the end of 8 years How long will it take to double?

6. If $5000 is invested in a savings account that pays 7.85% interest compounded continuously, how much money will be in the account after 12 yrs? How long will it take to double in value?

7. + Natural Logarithmic We call a log with a base of 10 “Common Log” We can call a log with a base of e “Natural Log” Natural Log is denoted with the “LN” All the same rules and properties apply to natural log as they do to regular logs

8. + Exponential to Log form 1. e x = 6 2. e x = 25 3. e x + 5 = 32

9. + Log to Exponential Form 1. Ln 1 = 0 2. Ln 9 = 2.197 3. Ln (5.28) = 1.6639

10. + Simplify 1. 3 Ln 5 2. Ln 5 + Ln 4 3. Ln 20 – Ln 10 1. 4 Ln x + Ln y – 2 Ln z

11. + Expand 1. Ln (xy 2 ) 1. Ln(x/4) 1. Ln(y/2x)

12. + Solving Exponential Equations 1. e x = 18 2. e x+1 = 30 3. e 2x = 12

13. + Solving Logarithmic Equations 1. Ln x = -2 2. Ln (2m + 3) = 8 3. 1.1 + Ln x 2 = 6

14. + Word Problems

15. Example 1: iPads The value of an iPad decreases at 35% per year. If the starting price of the iPad is $500, write the exponential function. How much will the iPad be worth after 5 years? When can you buy the iPad for $5?

16. Example 2: Forest Suppose the acreage of forest is decreasing by 2% per year because of development. If there are currently 4,500,000 acres of forest, how much forest land will there be in 6 years?

17. Example 3: Investing Find a bank account balance to the nearest dollar, if the account starts with $100, has an annual rate of 4%, and the money is left in the account for 12 years. If you wanted to buy a new gaming system for $250, when will you have enough?

18. + Example 4: Car Salesman Dave bought a car 8 years ago for $5400. To buy a similar car today would cost $12,500. Assuming a steady rate of increase, what was the yearly rate of inflation?

19. + Half Life Some unstable substances, like plutonium, decay over time. To measure the rate of decay, scientists refer to their “half life.” The half life is the time it takes for half the initial amount of the substance to decay.

20. Example 5: DDT The pesticide DDT was widely used in the United States until its ban in 1972. Write an equation that models the 15 year half-life of 100 grams of DDT. How much DDT would be remaining after 45 years?

21. Example 6: 228 Ac 228 Ac has a half life of 6.13 hours. Write an equation that models the half life of a 5 mg sample. How much 228 Ac would be remaining after one day?