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Properties of Logarithms They’re in Section 3.4a Proof of a Prop ‘o Logs Let x logb R In exponential form: and y logb S b R x b S y Let’s start with the product of R and S: RS b b x y RS b logb RS x y logb R logb S x y A Prop ‘o Logs!!! Properties of Logarithms Let b, R, and S be positive real numbers with b = 1, and c any real number. • Product Rule: logb RS logb R logb S • Quotient Rule: R log b log b R log b S S • Power Rule: log b R c log b R c Guided Practice Assuming x and y are positive, use properties of logarithms to write the given expression as a sum of logarithms or multiples of logarithms. log 8xy 4 log8 log x log y 4 log 2 log x log y 3 4 3log 2 log x 4 log y Guided Practice Assuming x is positive, use properties of logarithms to write the given expression as a sum or difference of logarithms or multiples of logarithms. x 5 x 5 ln ln x x 2 2 12 ln x 5 ln x 2 12 1 2 ln x 5 ln x 2 Guided Practice Assuming x and y are positive, use properties of logarithms to write the given expression as a single logarithm. 5ln x 2ln xy ln x ln xy 5 2 x x ln x ln x y ln 2 2 ln 2 x y y 5 5 2 2 3 Of the eight relationships suggested here, four are true and four are false (using values of x within the domains of both sides of the equations). Thinking about the properties of logarithms, make a prediction about the truth of each statement. Then test each with some specific numerical values for x. Finally, compare the graphs of the two sides of the equation. 1. ln x 2 ln x ln 2 2. log3 7 x 7log3 x 3. log 2 5x log 2 5 log 2 x x log x 5. log 4 log 4 7. log5 x log5 x log5 x 2 4. x ln ln x ln 5 5 6. log 4 x 3 3log 4 x 8. log 4x log 4 log x These four statements are TRUE!!! A few more problems… Assuming x and y are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms. log1000x log1000 log x 3 4 log x 4 3 ln 5 x y 2 4 ln x ln y 13 25 1 2 ln x ln y 3 5 And still a few more problems… Assuming x, y, and z are positive, use properties of logarithms to write the expression as a single logarithm. 4 1 y 4 4 log y log z log y log z log 2 z 3ln x yz 2 ln yz ln x y z ln y z ln x y z 3 2 2 9 3 6 2 4 9 5 10 Whiteboard… Write as a single logarithmic expression: log 100 log3x 5log x 2log 3x 5log x 2 log 3x log x 2 2 2 5 log 9 x log x 2 9x 9 log 10 log 8 x x 2 10 Let’s do an exploration… How do we evaluate 4 7 y log 4 7 ? Set equal to y: y log 4 7 First, switch to exponential form. ln 4 ln 7 Apply ln to both sides. y ln 4 ln 7 Use the power rule. We just proved the C.O.B.!!! ln 7 y 1.404 Divide by ln4. ln 4 y Change-of-Base Formula for Logarithms For positive real numbers a, b, and x with a = 1 and b = 1, a b a Because of our calculators, the two most common forms: log x log x log b log x ln x logb x log b ln b Guided Practice Evaluate each of the following. 1. 2. 3. ln16 log3 16 2.524 ln 3 1 log10 1.285 log 6 10 log 6 log 6 ln 2 ln 2 1 log1 2 2 ln 1 2 ln 2 Guided Practice Write the given expression using only natural logarithms. ln 3 x 1. log5 3x ln 5 2. log7 2x y ln 2 x y ln 7 Guided Practice Write the given expression using only common logarithms. log s 1. log 4 s log 4 2. log1 4 a 2b log a 2b log 1 4 log a 2b log 4 Graphs of Logarithmic Functions with Base b Rewrite the given function using the change-of-base formula. ln x 1 ln x g x logb x ln b ln b Every logarithmic function is a constant multiple of the natural logarithmic function!!! If b > 1, the graph of g(x) is a vertical stretch or shrink of the graph of the natural log function by a factor of 1/(ln b). If 0 < b < 1, a reflection across the x-axis is required as well. More Guided Practice Describe how to transform the graph of the natural logarithm function into the graph of the given function. Sketch the graph by hand and support your answer with a grapher. 1 1. g x log x ln x 5 ln 5 1 Vertical shrink by a factor 0.621 of approximately 0.621. ln 5 How does the graph look??? More Guided Practice Describe how to transform the graph of the natural logarithm function into the graph of the given function. Sketch the graph by hand and support your answer with a grapher. 1 ln x ln x 2. h x log x 14 ln 4 ln1 4 1 Reflect across x-axis, Vertical 0.721 shrink by a factor of 0.721 ln 4 How does the graph look???