INTERESTING QUESTIONS & “UNDOING” EXPONENTS: LOGS

1. INTERESTING QUESTIONS &“UNDOING” EXPONENTS: LOGS Hey, hey logarithm…When we get the blues

2. Recap Last week we looked at RATIONAL exponents and saw that • A square root is the same as an exponent of ½ • A cubed root is the exponent 1/3 • To evaluate powers with rational exponents, we “rip the exponent apart”. We also saw that radioactive materials will decay in an exponential fashion (half-life) We also saw that compound interest can be modeled using exponential equations

3. InterestInterest In general, the compound interest is • Where • A is the amount in the account at time, t • P is the principle (initial) amount • r is the decimal value of the interest rate • n is how many times per year the interest is compounded. Look for terms like: daily (n =365), semi-annually (n = 2), weekly (n = 52) and monthly (n =12)

4. InterestInterest Ex 1. A credit card charges 24.2% interest per year compounded monthly. There are $900 worth of purchases made on the card. Calculate the amount owing after 18 months. (Assume that no payments were made.)

5. InterestInterest Ex 2. An bank account earns interest compounded monthly. The investment doubles in 9.27 years. Calculate the annual interest rate.

6. A new type of question? 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? So this questions seems to be like all the others… And now we get Common bases

7. Solving for the Exponent We’re totally stuck! We presently have no way of solving for an exponent unless we can get common bases. exponent exponent So mathematicians invented logarithms. can also be written as base base argument argument We read this “log to base 2 of 8 equals 3.

8. Going from one form to another Write the following in logarithmic form. When the base is a ‘10’ we do not need to write it. This is because base 10 is what most calculators deal with.

9. Going from one form to another Write the following in exponential form.

10. Evaluating Logs By changing forms we can evaluate log expressions. Evaluate: a) This asks “2 to the what gives 32?” or We know this is 5. b) This asks “4 to the what gives 64?” or We know this is 3. c) This asks “1/4 to the what gives 32?” We can get common bases d) log 100 This asks “10 to the what gives 100”? We know this is 2.

11. Solving log equations: common bases To continue getting used to logs, we’ll look at these questions. Solve for x. We’re stuck in log form so go to exponential form a) We can solve this: x =1/8 We’re stuck in log form so go to exponential form b)

12. Solving log equations: common bases c) We’re stuck in log form so go to exponential form d) We’re stuck in log form so go to exponential form To solve for the base, we can undo the exponent by raising both sides to the 5/4. But wait… the base here is 10 Since the calculator uses base 10 I just type this in and get… log1.3=0.114

13. Try some… 1) Solve each equation a. a. b. c. d. e. f. g. h.

14. Try some… 1) Solve each equation b. a. b. c. d. e. f. g. h.

15. Try some… 1) Solve each equation c. a. b. c. d. e. f. g. h.

16. Try some… 1) Solve each equation d. a. b. c. d. e. f. g. h.

17. Try some… 1) Solve each equation e. a. b. c. d. e. f. g. h. Since I can’t get common bases, I’m stuck in exponential form. So I go to log form. My calculator can find this.

18. Try some… 1) Solve each equation f. a. b. c. d. e. f. g. h. I’m stuck in log form. So I go to exponential form. To solve for the base, I undo the exponent. I raise both sides to the -3/2 The negative in the exponent means I “flip the base”

19. Try some… 1) Solve each equation g. a. b. c. d. e. f. g. h. I’m stuck in log form. So I go to exponential form. Rip the exponent apart

20. Try some… 1) Solve each equation h. a. b. c. d. e. f. g. h. I don’t know what is. But I do notice that there’s a common base on both sides of the equation. Since the bases are equal, the ARGUMENTS must be equal.

21. So What? We can now write an equation where we solve for the exponent but how do we evaluate the exponent if we do not recognize the log or if we cannot get common bases? All of these started with the equation Are we any closer? Well we can write it in a log form Notice that the is equal to an exponent. It must also be an exponent. But that’s as far as we can get. How do the laws of exponents relate to logs?

22. The Laws of Logs Remember the laws of exponents: When multiplying powers with the same base, we keep the base and add the exponents. Let Go to log form If then If then When adding logs with the same base, we keep the log and base and multiply the arguments.

23. The Laws of Logs Remember the laws of exponents: When dividing powers with the same base, we keep the base and subtract the exponents. Let Go to log form If then If then When subtracting logs with the same base, we keep the log and base and divide the arguments.

24. Laws of Exponents Practice with the first two laws. Solve for x. I’m stuck in log form. So I go to exponential form.

25. The 3rd Law of Logs When we have a power of a power, we keep the base and multiply the exponents. Let Raise both sides to the exponent “b” The “down in front” rule! Ex. Evaluate If then Go to log form

26. 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

27. 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!

28. 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.

29. 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.

30. Lots o’ Logs Solve for x.