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.699... 6.68....70 6.6. Orer these logarithms from greatest to least: log 80, log 900, log 5000, log 5 0 000 Write each logarithm to base 0, then calculate its value. log 80 log 900 log 5000 log 5 0 000 log 80 log 900 log 5000 log 0 000 log log log log 5 6.9... 6.98... 6.8... 5.77... From greatest to least: log 80, log 900, log 5000, log 5 0 000 P DO NOT COPY. 5.6 Analyzing Logarithmic Functions Solutions
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 7.0790....079 So, 00 7.079 log 0.5 log a b log 0.609... 0.6 So, 0.6 6. 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 8 5.6 Analyzing Logarithmic Functions Solutions DO NOT COPY. P
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 0 6 8 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
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 0 6 8 (, 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 > 0. 5.6 Analyzing Logarithmic Functions Solutions DO NOT COPY. P
. 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 ( ). 8 6 0 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
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. 5 8 9 7 7 55.5 (, ) a 7, b (9, ) a, b The equation of the asymptote is. The omain of the function is <. 6 5.6 Analyzing Logarithmic Functions Solutions DO NOT COPY. P