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Understanding Natural Logarithms: The Relationship Between Exponents and Logs

Explore the fascinating world of natural logarithms and the constant "e" in this comprehensive guide. We delve into the relationship between the exponential function y = e^x and its inverse, the natural logarithmic function y = ln(x). You'll learn how to graph these functions, understand their inverse relationship, and apply logarithmic rules such as product, power, and quotient. Through practical examples involving solving equations with base "e" and real-life applications like continuously compounding interest, we'll build your confidence in using natural logs effectively.

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Understanding Natural Logarithms: The Relationship Between Exponents and Logs

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Presentation Transcript

  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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