Presentation Transcript

1. Logarithms:“undoing” exponents

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. Compound InterestFormula Where A is the $ amount in the account at time t (years)Pis the principle (initial) $ amount (when t = 0)iis the decimal value of the annual interest ratenis how many times per year the interest is compoundedt is the number of years Look for terms like:daily (n = 365),weekly (n = 52)quarterly (n = 4)semi-annually (n = 2) monthly (n = 12)
4. More Compound InterestExamples 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.) P = 900 i = 24.2% = 0.242 n = 12 t = 1.5 y = ?
5. More Compound InterestExamples Ex 2. 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… P = $400 i = 4.2% = 0.042 n = 12 t = ? y = $800 And now we geta common base… …except we can’t. …maybe by the end of class…
6. Logarithmic Form We`re totally stuck. So far we can only solve for an exponent when we can achieve common bases. Good thing we have logarithms to help. exponent exponent can also be written as base argument argument base That is “log base 2 of 8 equals 3” This form is helpful because the exponent value is no longer locked in the exponent position, in fact it is isolated.
7. Going from exponential to logarithmic form Write the following exponential equations in logarithmic form. When the base is 10, we usually leave it out… take a look at your calculator…
8. Going from logarithmic to exponential form Write the following logarithmic equations in exponential form.
9. Evaluating Logs By changing forms we can evaluate log expressions. Examples or, solve for x: a) Evaluate: This asks “2 to the what gives 32?” We know this is 5, so: b) Evaluate: or, solve for x: This asks “4 to the what gives 64?” We know this is 3, so:
10. Evaluating Logs c) Evaluate: or, solve for x: This asks “1/4 to the what gives 32?” We can get common bases : or, solve for x: d) Evaluate: We know this is 2, so:
11. Solving log equations Solving logarithmic equations takes some instinct, which only comes from practice, but to help you get you started, here is a flowchart with some possibly useful steps.
12. Solving log equations - Examples b) Solve for x. a) Solve for x. Unknown in the exponent Unknown in the argument
13. Solving log equations Unknown in the base c)
14. Solving log equations Unknown in the argument d) 10 But wait… the base here is... Since the calculator uses base 10, just type this in and get…
15. Solving log equations - Practice Solve each equation for x a. b. c. d. e. f. g. h.
16. Solving log equations - Practice Evaluations Solutions
17. Solving log equations - Practice Solutions Evaluations
18. Solving log equations - Practice Solutions Evaluations
19. Solving log equations - Practice Solutions
20. Solving log equations - Practice Solutions
21. Solving log equations - Practice Solutions Evaluations
22. Solving log equations - Practice Solutions We don’t know what is. But notice that there’s a common base on both sides of the equation. Since the bases are equal, the ARGUMENTS must be equal.
23. The first law of logarithms Remember the laws of exponents: First law of logarithms: When multiplying powers with the same base, we keep the base and add the exponents. When adding logs with the same base, we keep the log and base and multiply the arguments Let and So and
24. The second law of logarithms Second law of logarithms: Remember the laws of exponents: When subtracting logswith the same base, we keep the log and the base and dividethe exponents. When dividing powers with the same base, we keep the base and subtract the exponents. Let and So and
25. Laws of Logarithms - Practice Practice with the first two logarithm laws. Solve for x.
26. The third law of logarithms Third law of logarithms: The “down in front” rule Remember the laws of exponents: When we have a power of a power, we keep the base and multiply the exponents. If the argument of a logarithm is a power, the exponent can be moved “down in front”. Let So
27. Laws of Logarithms - Practice Practice with the third logarithm law. Solve. Evaluate.
28. Shortcut for evaluating logs This third law of logs is the key to evaluating logs with bases other than 10 on your calculator! Let change to exponential form take the log of both sides apply the “down in front” rule. remember that Logarithm Shortcut 1:To evaluate logab on your calculator, divide logb by loga. Remember,the base is on the bottom
29. Shortcut for solving for exponents This third law of logs is the key to solving exponential equations when common bases can’t be achieved! take the log of both sides apply the “down in front” rule. Logarithm Shortcut 2:To solve for an exponent, divide the log of the argument by the log of the base. Remember,the base is on the bottom
30. Laws of Logarithms - Practice Practice with the third logarithm law. Solve. Evaluate. Solve again. Evaluate again.
31. Laws of Logarithms – Practice Solve for x.
32. Another shortcut forevaluating certain logarithms logb(ba) = a Examples Challenge: Can you prove why this shortcut is true? logb(ba) = a
33. And back to the beginning… We can now solve this question! Ex 2. 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… P = $400 i = 4.2% = 0.042 n = 12 t = ? years y = $800 We can’t get common bases… but…..