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8.6 Natural Logarithms

8.6 Natural Logarithms. Natural Logs and “e”. The function y=e x has an inverse called the Natural Logarithmic Function. Start by graphing y=e x. 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!.

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8.6 Natural Logarithms

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  1. 8.6 Natural Logarithms

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

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

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

  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

  10. 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 = 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

  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

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