1. “BUT I STILL HAVEN’T FOUND WHAT I’M LOOKING FOR” -BONO Logs

2. Remember? Last class we had the problem: Ex 4. A bank account earns interest compounded monthly at an annual rate at 4.2%. Initially the investment was $400. When does it double in value? We got the equation… But then we couldn’t get common bases. So we introduced the log function. But we still haven't solved the problem!

3. Not yet… First we need to review logs This two equations are equivalent. And since logs are really just exponents, we have the laws of logs. 1) Multiplying arguments 2) Dividing arguments 3) Arguments with an exponent

4. And finally… Are we any closer to solving the original question? Let’s take the log of both sides… Now the “down in front” rule Divide by 12log1.0035 And my calculator can do this

5. A shortcut to the calculator rule So we have seen that can be written as So we do not need to take the log of both sides. We can go to log form And then write Remember that the base is on the bottom!

6. Lots o’ Logs Solve for x. Since I can’t get common bases, I’m stuck in exponential form. So I go to log form. Now I can use the calculator rule to change the base to 10.

7. Lots o’ Logs Solve for x. Since I can’t get common bases, I’m stuck in exponential form. So I go to log form.

8. Lots o’ Logs Solve for x. Since I can’t get common bases, I’m stuck in exponential form. So I go to log form.

9. Lots o’ Logs Solve for x.

10. Lots o’ Logs Solve for x. When there are 2 bases we take the log of both sides. Now move the exponents “down in front”

11. Lots o’ Logs Solve for x. We can get common bases!

12. Lots o’ Logs Solve these equations for x Stuck in log form so let’s write it in exponential form. Notice the common bases on the left hand side. Laws of logs apply: Stuck in log form so let’s write it in exponential form.

13. Lots o’ Logs

14. Back to Wrod There’s a problem with this word! A Sidney Crosby rookie card was purchased in 2005 for $15.oo. Its value is set to double every 2 years. When will the card be worth $90.00? We’ve set up equations like this before. Isolate the power! STUCK! Base is on the bottom In 5.17 years, the card is worth $90.

15. Back to Wrod A certain radioactive element has a half-life of 8.2 minutes. When will there be 1/10 th the original amount? We’ve set up equations like this before. Isolate the power! STUCK? No way! Base is on the bottom In 27.24 minutes only 1/10 th the original amount will remain. In this case y = (1/10)A o In this case y = (1/10)A o

16. Back to Wrod In Jan 2011, Ben and Josh opened saving accounts. Ben invested $250 in an account that will double every 8 years. Josh invested $600 in an account paying 6% interest per year compounded daily. When will they have the same amount of money? Ben’s Money Josh’s Money When they have the same amount, the y values are equal Now solve for ‘t’.

17. Back to Wrod Sarah bought a computer for $2000. Its value depreciates by 18% every two years. a. By what percentage does it depreciate every year? This means its value is 82% of the last value. The values create a pattern like this: X012 y20001811.081640 The CR is 0.906 so its losing 9.446% every year.

18. Back to Wrod Sarah bought a computer for $2000. Its value depreciates by 18% every two years. b. When is its value $99?

19. Back to Wrod In Jan 2011, Ben and Josh opened saving accounts. Ben invested $250 in an account that will double every 8 years. Josh invested $600 in an account paying 6% interest per year compounded daily. When will they have the same amount of money? We’re stuck in exponential form. With two bases, we’ll take the log of both sides. Notice the 365t is not the exponent all of the argument. So first, the laws of logs.

20. Back to Wrod In Jan 2011, Ben and Josh opened saving accounts. Ben invested $250 in an account that will double every 8 years. Josh invested $600 in an account paying 6% interest per year compounded daily. When will they have the same amount of money? We’re multiplying arguments so we can add their logs. Now the 365t can come down in front

21. Back to Wrod In Jan 2011, Ben and Josh opened saving accounts. Ben invested $250 in an account that will double every 8 years. Josh invested $600 in an account paying 6% interest per year compounded daily. When will they have the same amount of money? In 32.86 years, they will have the same amount.

22. Back to Wrod In Jan 2011, Ben and Josh opened saving accounts. Ben invested $250 in an account that will double every 8 years. Josh invested $600 in an account paying 6% interest per year compounded daily. When will they have the same amount of money? Ben’s Money Josh’s Money These are not exactly the same because we needed to round our values