Natural Logs and More Word Problems e is

+ Natural Logs and More Word Problems

+ e is a magic number! n https: //www. youtube. com/watch? v=UFgod 5 tm. LYY

+ 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.

+ Continuously Compounded Find the balance in an account paying 3. 2% annual interest on $10, 000 in 18 years compounded continuously.

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?

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?

+ 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

+ Exponential to Log form 1. ex = 6 2. ex = 25 3. ex + 5 = 32

+ Log to Exponential Form 1. Ln 1 = 0 2. Ln 9 = 2. 197 3. Ln (5. 28) = 1. 6639

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

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

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

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

+ Word Problems

Example 1: i. Pads y = a(1 + r)x The value of an i. Pad decreases at 35% per year. If the starting price of the i. Pad is $500, write the exponential function. How much will the i. Pad be worth after 5 years? When can you buy the i. Pad for $5?

Example 2: Forest y = a(1 + r)x 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?

Example 3: Investing y = a(1 + r)x 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?

+ Example 4: Car Salesman y = a(1 + r)x 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?

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

Example 5: DDT y = a(1 + r)x 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?

Example 6: 228 Ac y = a(1 + r)x 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?