Lesson 5.6 Exercises, pages

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1 Lesson 5.6 Eercises, pages 05 0 A. Approimate the value of each logarithm, to the nearest thousanth. a) log 9 b) log 00 Use the change of base formula to change the base of the logarithms to base 0. log 9 log 00 log 9 log log 00 log Orer these logarithms from greatest to least: log 80, log 900, log 5000, log Write each logarithm to base 0, then calculate its value. log 80 log 900 log 5000 log log 80 log 900 log 5000 log log log log log From greatest to least: log 80, log 900, log 5000, log P DO NOT COPY. 5.6 Analyzing Logarithmic Functions Solutions

2 B 5. Approimate the value of each logarithm, to the nearest thousanth. Write the relate eponential epression. a) log 7 00 b) log a b log 00 log 7 00 log So, log 0.5 log a b log So, a) Use technology to graph y log 5. Sketch the graph. Graph: y log log 5 b) Ientify the intercepts an the equation of the asymptote of the graph, an the omain an range of the function. From the graph, the -intercept is. There is no y-intercept. The equation of the asymptote is 0. The omain of the function is > 0. The range of the function is y ç. c) Choose the coorinates of two points on the graph. Multiply their -coorinates an a their y-coorinates. What o you notice about the new coorinates? Eplain the result. From the TABLE, two points on the graph have coorinates: (5, ) an (5, ) The prouct of the -coorinates is 5. The sum of the y-coorinates is. The new coorinates are (5, ), which is also a point on the graph. The logarithm of the prouct of two numbers is the sum of the logarithms of the numbers. 7. a) Use a graphing calculator to graph y log, y log, an y log 8. Sketch the graphs. log log Graph: y, y, an y log log log log Analyzing Logarithmic Functions Solutions DO NOT COPY. P

3 b) In part a, what happene to the graph of y log b, b > 0, b Z, as the base change? As b increases, from b, the graph of y log b is compresse log log b log vertically by a factor of:. log log b log 8. a) The graphs of a logarithmic function an its transformation image are shown. The functions are relate by translations, an corresponing points are inicate. Ientify the translations. A' From A to A, the translations are units left an unit up. The same translations relate B an B. y B' y g() B y f() A b) Given that f() log, what is g()? Justify your answer. After translations, the image of the graph of y log has equation: y k log ( h) Substitute: k an h The image graph has equation y log ( ); or y log ( ) So, g() log ( ) 9. a) How is the graph of y log ( 8) relate to the graph of y log? Sketch both graphs on the same gri. Compare y log ( ) with y k c log ( h): k 0, c,, an h Write y log ( 8) as y log ( ). The graph of this function is the image of the graph of y log after a vertical stretch by a factor of, a horizontal compression by a factor of, then a translation of units right. Use the general transformation: (, y) correspons to a h, cy kb 0 y y log ( 8) y log 6 8 The point (, y) on y log correspons to the point a, yb on y log ( ). (, y) a, yb (0.5, ) (.5, ) (, 0) (.5, 0) (, ) (5, ) (, ) (6, ) (8, ) (8, 6) P DO NOT COPY. 5.6 Analyzing Logarithmic Functions Solutions

4 b) Ientify the intercepts an the equation of the asymptote of the graph of y log ( 8), an the omain an range of the function. Use graphing technology to verify. From the table, the -intercept is.5. The equation of the asymptote is. The omain of the function is >. 0. a) Graph y = - log a b +. Compare y log a b with y k c log (h): k, c,, an h 0 Use the general transformation: ( ) y log y (, y) correspons to a h, cy kb The point (, y) on y log correspons to the point on y log a. b a, y b (, y) a, y b (0.5, ) (0.5,.5) (0.5, ) (,.5) (, 0) (, ) (, ) (, 0.75) (, ) (8, 0.5) b) Ientify the intercepts an the equation of the asymptote of the graph of y =- log a b +, an the omain an range of the function. Use the TABLE feature on a graphing calculator; the -intercept is. The equation of the asymptote is 0. The omain of the function is > Analyzing Logarithmic Functions Solutions DO NOT COPY. P

5 . Graph each function below, then ientify the intercepts an the equation of the asymptote of the graph, an the omain an range of the function. a) y log ( ) Compare y log ( ) with y k c log ( h): k, c,, an h Use the general transformation: (, y) correspons to a h, cy kb The point (, y) on y log correspons to the point (, y ) on y log ( ). y 0 (, y) (, y ) (0.5, ) (.75, ) (0.5, ) (.5, ) (, 0) (, ) (, ) (, ) (, ) (0, 5) (8, ) (, 6) y log ( ) 6 From the graph, the -intercept is approimately.9. From the table, the y-intercept is 5. The equation of the asymptote is. The omain of the function is >. b) y log ( ) Write y log ( ) as y log [( )]. Compare y log [( )] with y k c log ( h): k 0, c,, an h Use the general transformation: (, y) correspons to a h, cy kb The point (, y) on y log correspons to the point (, y) on y log ( ) y log ( ) y (, y) (, y) 8 (0.5, ) (.5, 8) (0.5, ) (.5, ) (, 0) (, 0) (, ) (5, ) (, ) (7, 8) From the table, the -intercept is. The equation of the asymptote is. The omain of the function is <. P DO NOT COPY. 5.6 Analyzing Logarithmic Functions Solutions 5

6 C. Graph the function y log ( ) 5, then ientify the intercepts, the equation of the asymptote, an the omain an range of the function. Write y log ( ) 5 as y 5 log [( )]. Compare y 5 log [( )] with y k c log ( h): y log ( ) 5 8 y 6 k 5, c,, an h Use the general transformation: (, y) correspons to a h, cy kb The point (, y) on y log correspons to the point a on y log ( ) 5., y 5b 6 0 (, y) a 9, b a, b (, 0) a, y 5b a 7 8, 7 b a 6, 6 b a 5, 5b To etermine the -intercept, solve the equation: 0 log ( ) 5 5 log ( ) 5 log ( ) Write in eponential form (, ) a 7, b (9, ) a, b The equation of the asymptote is. The omain of the function is < Analyzing Logarithmic Functions Solutions DO NOT COPY. P