SECTION 5.5 Properties of Logarithms 335 85. Mixed Practice If ( ) ( ) = = f x x g x e ln , ,x and ( ) = h x x ,2 find: (a) ( )( ) f g x . What is the domain of f g? (b) ( )( ) g f x . What is the domain of g f ? (c) ( )( ) f g 5 (d) ( )( ) f h x . What is the domain of f h? (e) ( )( ) f h e 86. Mixed Practice If ( ) = = f x x g x log , () 2,x 2 and ( ) = h x x4 , find: (a) ( )( ) f g x . What is the domain of f g? (b) ( )( ) g f x . What is the domain of g f ? (c) ( )( ) f g 3 (d) ( )( ) f h x . What is the domain of f h? (e) ( )( ) f h 8 Applications and Extensions In Problems 87–96, express y as a function of x. The constant C is a positive number. 87. = + y x C ln ln ln 88. ( ) = + y x C ln ln 89. ( ) = + + + y x x C ln ln ln 1 ln 90. ( ) = − + + y x x C ln 2ln ln 1 ln 91. = + y x C ln 3 ln 92. = − + y x C ln 2 ln 93. ( ) − = − + y x C ln 3 4 ln 94. ( ) + = + y x C ln 4 5 ln 95. ( ) ( ) = + − + + y x x C 3 ln 1 2 ln 2 1 1 3 ln 4 ln 96. ( ) = − + + + y x x C 2 ln 1 2 ln 1 3 ln 1 ln 2 97. Find the value of ⋅ ⋅ ⋅ ⋅ ⋅ log 3 log 4 log 5 log 6 log 7 log 8. 2 3 4 5 6 7 98. Find the value of ⋅ ⋅ log 4 log 6 log 8. 2 4 6 99. Find the value of ( ) ⋅ ⋅ ⋅ + ⋅ + n log 3 log 4 log 1 log 2. n n 2 3 1 100. Find the value of ⋅ ⋅ ⋅ ⋅ log2log4log8 log2. n 2 2 2 2 101. Show that ( ) ( ) + − + − − = x x x x log 1 log 1 0. a a 2 2 102. Show that ( ) ( ) + − + − − = x x x x log 1 log 1 0. a a 103. Show that ( ) ( ) + = + + − e x e ln 1 2 ln 1 . x x 2 2 104. Difference Quotient If ( ) = f x x log , a show that ( ) ( ) ( ) + − = + ≠ f x h f x h h x h log 1 , 0. a h 1 105. If ( ) = f x x log , a show that ( ) − = f x x log . a 1 106. If ( ) = f x x log , a show that ( ) ( ) ( ) = + f AB f A f B . 107. If ( ) = f x x log , a show that ( ) ( ) = − f x f x 1 . 108. If ( ) = f x x log , a show that α ( ) ( ) = α f x f x . 109. Show that ( ) = − M N M N log log log , a a a where a, M, and N are positive real numbers and ≠ a 1. 110. Show that ( ) = − N N log 1 log , a a where a and N are positive real numbers and ≠ a 1. 111. Challenge Problem Show that = b a log 1 log , a b where a and b are positive real numbers, ≠ a 1, and ≠ b 1. 112. Challenge Problem Show that = m m log log , a a 2 where a and m are positive real numbers and ≠ a 1. 113. Challenge Problem Show that = b m n b log log , a m a n where a, b, m, and n are positive real numbers, ≠ a 1, and ≠ b 1. 114. Challenge Problem Find n: ( ) ⋅ ⋅ ⋅ ⋅ + = … n log3 log4 log5 log 1 10 n 2 3 4 Explaining Concepts 115. Graph ( ) = Y x log 1 2 and ( ) = Y x 2 log 2 using a graphing utility. Are they equivalent? What might account for any differences in the two functions? 116. Write an example that illustrates why ( ) ≠ x r x log log . a r a 117. Write an example that illustrates why ( ) + ≠ + x y x y log log log 2 2 2 118. Does = − ( ) − 3 5? log 5 3 Why or why not? 123. Find the domain of ( ) = − − f x x 2 3 5 4. 124. Solve: + − < x 4 1 9 23 125. Find the vertex of ( ) =− + + f x x x 1 2 4 5, 2 and determine if the graph is concave up or concave down. 126. Find the center and radius of the circle − + + = x x y y 10 4 35 2 2 127. Find the average rate of change ( ) = f x x3 from −1 to 3. 128. Is the function ( ) = − f x x x x 5 3 2 4 3 even, odd, or neither? 119. Use a graphing utility to solve − − + = x x x 3 4 8 0. 3 2 Round answers to two decimal places. 120. Without solving, determine the character of the solution of the quadratic equation − + = x x 4 28 49 0 2 in the complex number system. 121. Find the real zeros of ( ) = + + + − − f x x x x x x 5 44 116 95 4 4 5 4 3 2 122. Graph ( ) = − f x x 2 using the techniques of shifting, compressing or stretching, and reflecting. State the domain and the range of f. Problems 119–128 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus. Retain Your Knowledge
RkJQdWJsaXNoZXIy NjM5ODQ=