1. Warm-Up 5/6 1. Find the value of 2. Solve. 2.3869 50,625

2. 11.6 Natural Logarithms Natural logarithms are just like the logarithms we’ve been using in previous sections. The only difference is they are in base “e”. Whenever we have an equation with “e” we will use the natural logarithm They are written: ln x = log e x What is ln e = ? e ^ ? = e ………..1 st power Therefore: ln e = 1 What is ln 1 = ?Think about it……… 0 “log base b to number 1 is always 0 just for fun”

3. Easy stuff:type into calculator ex1: evaluate ln0.0089 -4.7217 Ex2: evaluate antiln(-0.7831) 0.4570 You probably won’t see “antiln” too often, it makes more sense to write it how?? e x

4. Ex3: solve: 7.2=-28.8 lnx Divide by -28.8 Re-write in exponential form Plug into calc and solve

5. Ex: 12e x = 108 Always try to get the “exponential term” alone first (divide by 12) Now take the ln of both sides solve e x = 9 lne x = ln9 xlne = ln9 x=2.20

6. Ex4: solve using natural logs 5 2x = 7 x+1 Take the natural log of both sides Use your log properties “ finding answers in the hunt powers of logs can go up front ” Divide by the natural log 1 st, either side, you pick Solve the problem, no logs anymore

7. Ex 5: solve 4.5 > e 0.031t Take natural log of both sides Use properties Remember lne = 1 Solve away divide by.031 1 t < 48.5186

8. Assignment 11.6 p. 736 #18 – 43