1. 8.6 Natural Logarithms

2. Natural Logs and “e” Start by graphing y=e x The function y=e x has an inverse called the Natural Logarithmic Function. Y=ln x

3. What do you notice about the graphs of y=e x and y=ln x? y=e x and y=ln x are inverses of each other! We can use the natural log to “undo” the function y= e x (and vice versa).

4. All the rules still apply You can use your product, power and quotient rules for natural logs just like you do for regular logs Let’s try one:

5. Solving with base “e” 2. Divide both sides by 7 3. Take the natural log of both sides. 4. Simplify. 1. Subtract 2.5 from both sides 5. Divide both sides by 2 x = 0.458 6. Calculator

6. Another Example: Solving with base “e” 1. Take the natural log of both sides. 2. Simplify. 3. Subtract 1 from both sides x = 2.401 4. Calculator

7. Solving a natural log problem 2. Use a calculator 3. Simplify. 1. Rewrite in exponential form To “undo” a natural log, we use “e”

8. Another Example: Solving a natural log problem 1. Rewrite in exponential form. 2. Calculator. 3. Take the square root of each time 3x+5 = 7.39 or -7.39 4. Calculator X=0.797 or -4.130 5. Simplify

9. Let’s try some

11. 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

12. Going back to our continuously compounding interest problems... $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