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IB Math SL Year 2 Name: ________________________ Date: _________________________ . 2-2: Laws of Logs, Log Rules, Log Equations Key Notes 1. Laws of Logs and Change of Base 2. Solving Equations with Logs ο· What do I need to know? Definitions for: o Exponent o Power o Base o Log ο· Calculation for/Process: o Perform Log Rules o Change bases o Rewriting Logs to equivalent expressions ο· Definitions for: o Power/exponent ο· Calculation for/Process: o Solving equations with logs Notes to Self In this lesson we will revisit the following learning goals: 1. 2. 3. 4. What are the laws of logs? How do we change the base of logs? How do we solve equations where the variable is in the exponent? What methods do we use to solve equations that involve logs? Station 1: Logarithms and Log rules We can re-write Exponents as Logarithms (A logarithm is the ______________ to which a number must be raised in order to get some other number) If π = π π₯ then log π π = π₯ Examples) ππ = ππ is equivalent to π₯π¨π π ππ = π ππ = π is equivalent to π₯π¨π π π = π ππ = π is equivalent to π₯π¨π π π = π IB Math SL Year 2 Letβs try it! Example My thinking βThe log is the number I must raise the 5 to in order to get 625.β Evaluate log 5 625 1 Evaluate log 5 6252 Without a calculator: Evaluate log10 10 Solution 1 ο· Simplify 6252 ο· βThe log is the number I must raise the 5 to in order to get _________.β βThe log is the number I must raise the 10 to in order to get 10.β **SO, BIG TAKE AWAY: When base and number match, the log = 1. Without a calculator: Evaluate log 2 1 βThe log is the number I must raise the 2 to in order to get 1.β **SO, BIG TAKE AWAY: When number = 1, the log = 0. This is because anything to the 0-power is 1. *Log rules: Remember- A logarithm is an ____________________. Similar laws apply to logs that hold for exponents. When we multiply expressions of like bases we add exponents ππ × ππ = ππ+π Using what we know about exponents: Multiplyβ¦ When we divide expressions of like bases we subtract exponents ππ = ππβπ ππ log π₯π¦ =________________________ Divideβ¦ When we raise an expression to a power we multiply exponents (ππ )π = πππ Important facts to memorizeβ¦ π₯ log π¦ =_____________________ Powerβ¦ log π₯ π = π log π₯ *This is on page _______ in our formula booklets! IB Math SL Year 2 Letβs try it! Example Express as a single logarithm: log 5 + log 6 My thinking βWhen adding logs with the same base, we can multiply the big numbers and take the log of that answer.β Express as a single logarithm: ο· Simplify/condense the coefficient as an exponent. ο· βWhen subracting logs with the same base, we can divide the big numbers and take the log of that answer.β Solution 3log x - 2log y Station 2: Change of base - Sometimes you need to change the base of a logarithm and there is a formula that enables you to do this. - We most often find ourselves using the formula to change logarithms into base 10 so we can then use the log button on our calculator. *This is on page _______ in our formula booklets! Letβs look at some: Example Use the change of base formula to evaluate log 4 9 to 3 significant figures. My thinking βHere we have to use change of base because our calculator can only do logarithms of base 10.β log π₯ 3 = π and log π₯ 6 = π. Find log 3 6 in terms of π and π. ο· Use the change of base formula to 1 evaluate log 5 7 ο· Evaluate each piece (replace with a and b) Use change of base formula. Solution IB Math SL Year 2 Station 3: SOLVING EQUATIONS Letβs see what we already know how to do: 1. Without a calculator: Solve for x: 2π₯ = 32 2. Without a calculator: Solve for the exact value of x: 2π₯ = 30 *What is the problem we encounter? __________________________________________________________ Letβs Try Three Togetherβ¦ 1. Solve log π (π₯ 2 ) = log π (3π₯ + 4) 2. Solve ln(12 β π₯) = ln π₯ + ln(π₯ β 5) IB Math SL Year 2 Station 4: Mixed Practice (to be completed and checked with answer key) 1. Write these expressions in log form. a. X = 29 b. x = ab 2. Write these expressions in exponent form: a. X = πππ2 8 b. X = ππππ π 3. Evaluate: a. log 2 25 b. log 64 4 1 81 π. log 5 β5 d. log 3 4. Without a calculator: a. log 6 6 b. log 8 1 5. Find the value of π₯ if log 2 π₯ = 5 1 2 6. Express log 2 5 + log 2 36 β log 2 10 as a single logarithm. IB Math SL Year 2 βπ₯ 7. Exam Style: Given that π = log 5 π₯, π = log 5 π¦, and π = log 5 π§, write log 5 (π¦2 π§3 ) in terms of π, π, and π. 8. Rewrite as a single logarithm: a. log24 β log2 b. log x - log y β log z 9. Solve: a. log 4 π₯ = 3 b. log π₯ 64 = 2 10. Without a calculator: a. log π₯ = 2 b. log x = -1 c. log x = 0 d. log π π 11. Use a calculator to a. evaluate π₯π¨π π to 3 decimal places. b. Solve these equations, giving your answers to 3 significant figures. a. log π₯ = 3 b. 10π₯ = 0.75 12. Write these expression in the form a + blogx, where a and b are integers. In other words, rewrite each log expression below using log rules. IB Math SL Year 2 13. 14. Use the change of base formula to evaluate these expressions to 3 significant figures. a. πππ2 7 b. πππ3 (0.7) 15. Solve for x: 8π₯ β 5(4π₯ ) = 0 IB Math SL Year 2 16. Solve these equations for x: a. πππ9 (π₯ β 2) = 2 b. πππ1 (3 β π₯) = 5 2 17. Given that πππ2 π₯ + πππ2 (2π₯ + 7) = πππ2 π΄, find an expression for A in terms of x. Hence or otherwise solve πππ2 π₯ + πππ2 (2π₯ + 7) = 2. Now, go back through your notes and highlight the questions you need to go over with a peer or in extra help!!