8.6 Natural Logarithms

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

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).

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:

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 = Calculator

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

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

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 Calculator X=0.797 or Simplify

Let’s try some

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

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