Week 10 Properties of Logarithms Exponential Equations Assignment

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Properties of Logarithms; Logarithmic and Exponential Equations From last week (5.4) Post 1 https://mediaplayer.pearsoncmg.com/assets/bIl00_lvtNkCqnHF077MHAI55lbv7h_y Author in Action: Solve Logarithmic Equations (6:43) Solving Basic Logarithmic Equations When solving simple logarithmic equations (they will get more complicated in Section 5.6) follow these steps: 1. Isolate the logarithm if possible. 2. Change the logarithm to exponential form and use the strategies learned in Section 5.3 to solve for the unknown variable. Example 7*: Solve Logarithmic Equations Solve the following logarithmic equations (a)* log2 ( 2x +1) = 3 (b)* log x 343 = 3 End of Post 1 (c) 6 − log(𝑥) = 3 (d) ln ( x ) = 2 (e) 7 log 6 (4 x) + 5 = −2 (f) log 6 36 = 5 x + 3 Copyright © 2016 Pearson Education, Inc. Section 5.5 & 5.6 Steps for solving exponential equations of base e or base 10 1. Isolate the exponential part 2. Change the exponent into a logarithm. 3. Use either the “log” key (if log base 10) or the “ln” (if log base e) key to evaluate the variable. Example 8*: Using Logarithms to Solve Exponential Equations Solve each exponential equation. (a) e x = 7 (b)* 2e3 x = 6 (c) e5 x −1 = 9 Copyright © 2016 Pearson Education, Inc. Properties of Logarithms; Logarithmic and Exponential Equations Section 5.5: Properties of Logarithms https://mediaplayer.pearsoncmg.com/assets/RvacGa5_033bZykLJ_5DvHQt1seryy2z Author in Action: Work with Properties of Logarithms (12:32) Exploration 1: Establish Properties of Logarithms Calculate the following: (a) log5 (1) (b) log 2 (1) (c) log(1) (d) ln(1) (e) log 5 (5) (h) ln(e) (f) log 2 (2) (g) log(10) Properties of Logarithms: To summarize: 1. log a 1 = _______ 2. log a a = _______ Exploration 2: Establish Properties of Logarithms In section 5.4, we found that the inverse of the function f ( x) = log 2 ( x) was f −1 ( x) = 2x . In fact, in general we can say that the functions defined by g ( x) = log a ( x) and h( x) = a x are inverse functions. Knowing what you know about inverse functions, evaluate: (a) g (h(r )) (b) h( g (m)) Properties of Logarithms: To summarize: In the following properties, M and a are positive real numbers, where a  1 , and r is any real number : 3. log a a r = _______ 4. a log a M = _______ Exploration 3: Establish Properties of Logarithms Show that the following are true  1000  (a) log (100 10 ) = log(100) + log(10) (b) log   = log(1000) − log(100)  100  (c) log103 = 3log(10) Copyright © 2016 Pearson Education, Inc. Section 5.5 & 5.6 Properties of Logarithms: To summarize: In the following properties, M, N, and a are positive real numbers, where a  1 , and r is any real number : M 5. loga ( MN ) = __________ 6. log a  N  r  = __________ 7. log a M = ________  Post 2 Example 1*: Work with the Properties of Logarithms Use the laws of logarithms to simplify the following: æ1ö (c) log 1 ç ÷ è ø 2 2 20 (a) 3log3 18 (b) 2log2 ( −5) End of Post 2 Example 2: Work with the Properties of Logarithms Use the laws of logarithms to find the exact value without a calculator. (a) log3 (24) − log3 (8) (b) log8 (2) − log8 (32) (c) 6log6 (3) + log6 (5) (d) e log e2 (25) Copyright © 2016 Pearson Education, Inc. (d) ln(e3 ) Properties of Logarithms; Logarithmic and Exponential Equations https://mediaplayer.pearsoncmg.com/assets/odn0bdpcDuXkrxTGFf6K7rb14h6pIgDQ Author in Action: Write a Logarithmic Expression as a Sum or Difference of Logarithms (9:47) Post 3 Example 3*: Write a Logarithmic Expression as a Sum or Difference of Logarithms Write each expression as a as a sum or difference of logarithms. Express all powers as factors.  x2 y3  2 (a) log3 ( x − 1)( x + 2 )  , x  1 (b) log5      z  End of Post 3 https://mediaplayer.pearsoncmg.com/assets/S3NHo0fz5nvLVdbobAY_skodcxtlzNdN Author in Action: Write a Logarithmic Expression as a Single Logarithm (5:47) Post 4 Example 4*: Write a Logarithmic Expression as a Single Logarithm Write each of the following as a single logarithm. (a) log2 x + log2 ( x − 3) (b) 3log6 z − 2log6 y 1 (c) ln ( x − 2 ) + ln x − 5ln ( x + 3) 2 End of Post 4 https://mediaplayer.pearsoncmg.com/assets/OTEWo0md3ccEoWXNl49RPD7wnUNLS1nY Author in Action: Evaluate a Logarithm Whose Base Is Neither 10 Nor e (9:20) Properties of Logarithms continued: In the following properties, M, N, and a are positive real numbers where a  1 : 8. If M = N, then ___________________ 9. If log a M = log a N , then ___________ Let a ≠ 1, and b ≠ 1 be positive real numbers. Then the change of base formula says: 10. log a M = _____________ Copyright © 2016 Pearson Education, Inc. Section 5.5 & 5.6 Why would we want to use the change of base formula? Post 5 Example 5*: Evaluate a Logarithm Whose Base is Neither 10 nor e. Approximate the following. Round your answers to four decimal places. (a) log3 12 (b) log7 325 End of Post 5 Summary Properties of Logarithms: In the following properties, M, N, and a are positive real numbers, where a  1 , and r is any real number : log a 1 = _______ log a a = _______ log a M r = _______ a loga M = _______ log a a r = _______ a r = _______ M log a  N If M = N, then ___________________ If loga ( MN ) = ______________   = ______________  log a M = log a N , then ___________ Change of base formula: log a M = _____________ Copyright © 2016 Pearson Education, Inc. Properties of Logarithms; Logarithmic and Exponential Equations Section 5.6: Logarithmic and Exponential Equations We will use the properties of logarithms found in Section 5.5 to solve all types of equations where a variable is an exponent. The following definition and properties that we’ve seen in previous sections will be particularly useful and provided here for your review: Summary: Know this! The logarithmic function to the base a, where a  0 and a  1 , is denoted by y = log a x (read as “y is the logarithm to the base a of x”) and is defined by: y = log a x if and only if x = a y The domain of the logarithmic function y = logax is x > 0. Properties of Logarithms: In the following properties, M, N, and a are positive real numbers, where a  1 , and r is any real number : 1. a log a M = M 2. log a a r = r 3. loga ( MN ) = log a M + log a N M  4. log a   = log a M − log a N N  5. log a M r = r log a M 6. a x = e x ln a 7. If M = N , then loga M = loga N 8. If loga M = loga N , then M = N . Strategy for Solving Logarithmic Equations Algebraically 1. Rewrite the equation using properties of logarithms so that it is written in one of the following two ways: log a x = c or log a (something) = log a (something else) . 2. If the equation is of the form log a x = c change it to exponential form to undo the logarithm and solve for x. 3. If the equation is of the form log a (something) = log a (something else) use property 8 to get rid of the logarithms and solve. 4. Check your solutions. Remember that The domain of the logarithmic function y = logax is x > 0. Copyright © 2016 Pearson Education, Inc. Section 5.5 & 5.6 Copyright © 2016 Pearson Education, Inc. Properties of Logarithms; Logarithmic and Exponential Equations Example 1: Solve Logarithmic Equations Solve the following equations: (a) log3 4 = 2log3 x (b) log2 ( x + 2) + log2 (1 − x ) = 1 Example 2: Solve Logarithmic Equations Solve the following equations: (a) ln ( x −1) + ln x = ln ( x + 2) (c) log(1 − c) = 1 + log(1 + c) (b) log 4 (h + 3) − log 4 (2 − h) = 1 (d) ln(3m + 1) = 2 + ln(m − 3) Copyright © 2016 Pearson Education, Inc. Section 5.5 & 5.6 https://mediaplayer.pearsoncmg.com/assets/RLuI3B5OdEiaUCkxqTSIUURkVZfzpARF Author in Action: Solve Exponential Equations (20:52) Example 3*: Solve Exponential Equations Solve the following equations: (a) 9 x − 3x − 6 = 0 (b) 3 x = 7 (c) 5  2 x = 3 (d) 2 x −1 = 52 x +3 Copyright © 2016 Pearson Education, Inc. Properties of Logarithms; Logarithmic and Exponential Equations https://mediaplayer.pearsoncmg.com/assets/IvCHjWCwfBRiMPsNHRYFBYi5X3__YvK9 Solve Logarithmic and Exponential Equations Using a Graphing Utility (2:32) So far we have solved exponential and logarithmic equations algebraically. Another method we can use is to solve by graphing. Here is a list of steps for how to do this: Solving by Graphing 1. 2. 3. 4. Put one side of the equation in y1 . Put one side of the equation in y2 . Graph the equations and find the point at which they intersect. The x value is your solution. Example 4*: Solving Logarithmic and Exponential Equations Using a Graphing Utility Solve e x = − x using a graphing utility. Copyright © 2016 Pearson Education, Inc. ...
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