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Math 110 Lecture #19 CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations. Change-of-Base Formula. For any logarithmic bases a and b, and any positive number M, log M=logaM b log b a Problem #1. Use your calculator to find the following logarithms. Show your work with Change-of-Base Formula. log 10 log 9 log 11 a) b) c) 2 1 7 3 Using the Change-of-Base Formula, we can graph Logarithmic Functions with an arbitrary base. Example: log x = ln x 2 ln2 logx log2 x = log2 y = log2 x 1 Math 110 Lecture #19 CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations. Properties of Logarithms. If b, M, and N are positive real numbers, b ≠1, p, x are real numbers, then 1. log MNM=+log log N product rule bbb 2. log M =−log M log N quotient rule bbb N log Mp = plog M 3. power rule bb ⎧ x 4. ⎪logbbx= inverse property of logarithms ⎨ log x b bx=>,0x ⎪ ⎩ 5. log M ==log N if and only if M N bb . This property is the base for solving Logarithmic Equations in form log gx=log hx. ( ) ( ) bb Properties 1-3 may be used for Expanding and Condensing Logarithmic expressions. Expanding and Condensing Logarithmic expressions. 2 Math 110 Lecture #19 CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations. Problem #2. Express each of the following expressions as a single logarithm whose coefficient is equal to 1. a) 1⎡⎤ ++ −− 3log xx1 2log 3 log7 ()() 5⎣⎦ 11 ⎡⎤ ++ −+ b) ln xx1 2ln 1 ln x ()() ⎣⎦ 23 11 ⎡⎤ c) +− + + ln xx3 ln 3ln x1 () () ⎣⎦ 25 1⎡⎤ −+ +− d) log xx2 2log 2 log5 ()() 2⎣⎦ Problem #3. Expand a much as possible each of the following. x2y a) log z5 x3y b) ln 4 z3 Solving Logarithmic Equations. 3 Math 110 Lecture #19 CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations. 1. Solving the Simplest Logarithmic Equation (SLE). Given: log xa= a b , b > 0, b ≠1, is any real number. According the definition of the logarithm this equation is equivalent to x = ba. 2. According to properties of logarithms, if log M = log N , then M = N. bb Remember, check is part of solution for Logarithmic Equations. Problem #4. Solve the following Logarithmic Equations. a) log2 x =5 b) log x−=25 3 () c) 2 log xx−=log6 () d) log x+=4 −3 1 () 2 4
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