1. Warm up 1.Evaluate the expression log 752.6. 2.Find the value of using the change of base formula. 3.Solve the equation

2. Lesson 11-6 Natural Logarithms Objective: To find natural logarithms of numbers To solve equations and inequalities using natural logarithms To solve real-world applications with natural logarithmic functions

3. All the properties that apply to common logs also apply to natural logs Converting a common log to a natural log: use change of base formula. Example:

4. Antiln The antilog for natural logs is antiln. Antiln is taking the term to e x

5. Solving a natural log problem 2. Use a calculator 3. Simplify. 1. Rewrite in exponential form take the antiln of both sides antilnantiln 3

6. Solve

7. Convert to natural logarithim

8. Solve:

9. Going back to our continuously compounding interest problems... A $20,000 investment appreciates 10% each year. How long until the stock is worth $50,000? Remember our base formula is A = Pe rt... We now have the ability to solve for t A = $50,000 (how much the car will be worth after the depreciation) P = $20,000 (initial value) r = 0.10 t = time From what we have learned, try solving for time

10. $20,000 appreciates 10% each year. How long until the stock is worth $50,000? A = $50,000 (how much the car will be worth after the depreciation) P = $20,000 (initial value) r = 0.10 t = time

11. Practice How long does it take an amount of money to double if it is invested at a rate of 8% per year compounded continuously?