Natural Logarithms

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

Look at the graph below - what relationship do the two functions have? then

We can use the natural log to “undo” the function y= ex (and vice versa). y=ex and y=ln x are inverses of each other!

Using the natural log - ln Without using a calculator find the value of: Use a calculator to find: = 3 =0 = 4 =1 = =2 = -3 = = -1 = n

The laws of natural logarithms

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

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

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

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

Let’s try some

Let’s try some

Remember the base of a natural log is e. Find x, if Find x, if Remember the base of a natural log is e. Take a natural log of both sides. Rearrange in index form. Use the power rule. Find x in each of the following: Find x in each of the following:

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 = Pert . . . 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

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