Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is give 7. 8. 0,,6, 6 0,,4, 6 Write each equatio i its equivalet epoetial form. 9. 6 log 64 0. log9. log y. Write each equatio i its equivalet logarithmic form. 4 6. 4. 8 y 00 Simplify the followig epressios.. log7 49 6. log 7 7. log 8. log 9 9. log6 6 0. log 7. log89. log6. 7 4. loglog. log log log 8 4 log 6 6 6. log 8 log log 8 log.00 0
7. log 6 log 9 8. log6 4 log7 9. log7 log8 log 0. log log 4 7 Graph the followig logarithmic fuctios by modifyig the graph of f log domai of each fuctio. g log.. g log. g log 4. g log 40 64 64 Epad each epressio as much as possible, ad simplify wheever possible.. log 64 7 6. log 9 7. 9. 8 7 log b 40. 9 8 log0 M 7 4.. Fid the 9 log0 0,000 8. log9 log 4. log b y 4. log 44. log 00 0 7 Compress each epressio as much as possible, ad simplify wheever possible. log 0 log 4 46. log40 log 47. log 7log 4. 0 0 y 48. 7log0 log0 y 49. log 4 log 4 y log 4 Use a calculator to evaluate the followig logarithms to four places. 0. log67. log6 7.. log 400. Simplify 0 0 0 log 8 log 4. Fid the value of for 0, for 0. a so that log log log a. 8 6. Solve the followig epoetial equatios. 7 7. 8. 6 7 7 9. 8 4 60. 4 4 0 6. 4 0
6. log Solve the followig logarithmic equatios. 6. log 64. log 4 6. log log 66. 6 6 7 log log 67. log 4 log 9 68. log 7 log log 7 0 0 0 log log log 69. 70. log 8 log 7 log 8 7. 4 Solve the followig mied equatios. 7. log 6 log log 0 7. 7. 9 74. 76. 9 8 6 4 77. log 8 0 78. log log 79. For what values of b is f log b a icreasig fuctio? a decreasig fuctio? 80. For ab, 0,, ab, fid the value of log b log a a. 8. Fid values of a ad b so that the graph of f b ae cotais the poits 6,0. 8. Suppose that is a umber such that 4. Fid the value of b.,0 ad 8. Suppose that is a umber such that. Fid the value of 4. 84. Suppose that is a umber such that. Fid the value of 8. 8. Suppose that is a umber such that. Fid the value of 86. For by, 0 ad b,, show that log b logb y y. log b 9.
87. f g 6 Fid a simplified formula for f log, 6 g for the give fuctios. 88. f g log, 89. f 6, g log 6 90. f, g log Whe solvig iequalities, it is commo to apply a fuctio to all sides of the iequality. If the fuctio is a icreasig fuctio, the the directio of the iequality is preserved, but if the fuctio is a decreasig fuctio, the the directio of the iequality is reversed. Here s why: If f is icreasig, the for y, f f y, ad you see that the iequality directio is preserved. If f is decreasig, the for y, f f y, ad you see that the iequality directio is reversed. Here are some eamples: a) To solve the iequality 0 f, you apply the fuctio to both sides of the iequality. Sice it s a icreasig fuctio, the directio of the iequality is preserved, ad you get. b) To solve the iequality 0 f, you apply the fuctio to both sides of the iequality. Sice it s a decreasig fuctio, the directio of the iequality is reversed, ad you get. c) To solve the iequality, you apply the fuctio f to all sides of the iequality. Sice it s a icreasig fuctio, the directio of the iequality is preserved, ad you get 6. d) To solve the iequality, you apply the fuctio f to all sides of the iequality. Sice it s a decreasig fuctio, the directio of the iequality is reversed, ad you get 6 6. e) To solve the iequality 0 f, you apply the fuctio 0 log to all sides of the iequality. Sice it s a icreasig fuctio, the directio of the iequality is preserved, ad you get log0 log0 0 log0.
f) To solve the iequality, you apply the fuctio f log to all sides of the iequality. Sice it s a decreasig fuctio, the directio of the iequality is reversed, ad you get log log log 0. 9. 4 6 Use the previous discussio to solve the followig iequalities: 9. log 0 9. 4 94. 9. 96. log 0 8 4 6 97. 98. log 4 99. Simplify the followig as much as possible: log log log log A B A B 00. 0. log ABC log ABC log A log B log A log AB 0. log AB 0 0. 00 log Alog B 999,999 04. Fid the eact value of the sum log log log 4 log,000,000.,000,000 0. Fid the eact value of the sum log 4 log log log 999,999. If all the terms of a sequece are positive, it is sometimes coveiet to aalyze the related sequece. If, for N If, for N, the the sequece, the the sequece is evetually odecreasig. is evetually icreasig. If, for N,
the the sequece is evetually oicreasig. If, for N, the the sequece is evetually decreasig. Test the followig sequeces by aalyzig. 06. 07. 08. 4 09..! 0.. 00 4. 4 4. 4 4. Fid a eplicit formula for a i the followig recursively defied sequece: a, a a. {Hit:, so a, a, a 4 4 formulate the patter.} a, 4 6. A sequece a is give recursively by,, see if you ca 4 4 a a. If 4 a, the what s a? 7. Fid so that,, are cosecutive terms of a arithmetic sequece. 8. Fid so that,, are cosecutive terms of a arithmetic sequece. 9. How may terms must be added i a arithmetic sequece whose first term is ad whose commo differece is to get the sum 09? 0. How may terms must be added i a arithmetic sequece whose first term is 78 ad whose commo differece is -4 to get the sum 70?. a) Determie the commo differece for the followig arithmetic sequece:,,7,,9,,
b) Fid a formula for a that geerates the umber i this sequece. c) Is 7 a umber i this arithmetic sequece?. a) Determie the commo differece for the followig arithmetic sequece: 8,6,4,,0,, b) Fid a formula for a that geerates the umbers i this arithmetic sequece. c) What is the 99 th umber i this arithmetic sequece?. Fid so that,, are cosecutive terms of a geometric sequece. 4. Fid so that,, are cosecutive terms of a geometric sequece.. a) Determie the commo ratio for the followig geometric sequece: 8,4,,,,, 4 b) Fid a formula for a that geerates the umbers i this geometric sequece. c) Is 04 a umber i this geometric sequece? 6. a) Determie the commo ratio for the followig geometric sequece: 7.,,4, 8,6,, b) Fid a formula for a that geerates the umbers i this geometric sequece. c) Is a umber i this geometric sequece? Determie if the followig geometric series coverge or diverge. If it coverges, fid its sum. 4 8 8. 9 6 9. 9 7 4 6 64 0. 9 6 64 k. 8 k. k k k. 4 4. 4 k k k
Prove the followig usig the Priciple of Mathematical Iductio: 9 4. 6. 7 7. 4 7 8. 9. 4 40. 7 4. 4. 4 6 4. is divisible by 44. is divisible by 6 4. 4 4 4 46. 47. 48. is divisible by 49. 7 4 is divisible by. 0.!!!!. 4
. is divisible by 8.. 9 4 is divisible by. 4. 6 0 4 4!!. k j for some pair of atural umbers, k ad j, for 8. ;. 7. is divisible by, for 0. 8. ; 4. 6. 4 9 6 9. 0 0 is divisible by 8, for 0. Fid a formula for the th term of the followig sequeces: 60. 4 8,,, 6.,,,, 6.,6,,4, 48, 9 7 6. a 4, a.a 64. a a a 6. a a a, 4, 66. a, a a a 67. a, a 4 68. I a geometric sequece of real umber terms, the first term is ad the fourth term is 4. Fid the commo ratio. 69. Fid the seveth term of a geometric sequece whose third term is 9 4 ad whose fifth term is 8 64. 70. For what value(s) of k will k 4, k,k form a geometric sequece? Fid the sum of the followig series: 7. j 7. 9 7 74. 7. j e e e e 7. 4 76. 0 4
77. 78. 80. 0 79. 0 0 0 8. e 8. 0 0 cos 8. Solve the equatio i 6 for. i 84. Fid the secod term of a arithmetic sequece whose first term is ad whose first, third, ad seveth terms form a geometric sequece. 8. The figure shows the first four of a ifiite sequece of squares. The outermost square has a area of 4, ad each of the other squares is obtaied by joiig the midpoits of the sides of the square before it. Fid the sum of the areas of all the squares.
86. The equatio e has 0 as its oly solutio. If the Method of Successive Approimatios is applied to approimate this solutio, graphically idicate the result if the startig guess is a) a positive umber b) a egative umber
87. The equatio has 0 ad as its solutios. If the Method of Successive Approimatios is applied to approimate these solutios, graphically idicate the result if the startig guess is a) greater tha b) betwee 0 ad
c) less tha 0 88. Cosider the statemet. a) Show that if the statemet is true for k, the it must be true for k. b) For which atural umbers is the statemet true? b 89. If log b b, the what s the value of b? 90. The third term of a arithmetic sequece is 0. Fid the sum of the first terms. 9. The third term of a geometric sequece is 4. Fid the product of the first terms. 9. Is there a geometric sequece cotaiig the terms 7, 8, ad i ay order ad ot ecessarily cosecutive? If so, give a eample. If ot, show why. 9. Is there a geometric sequece cotaiig the terms,, ad i ay order ad ot ecessarily cosecutive? If so, give a eample. If ot, show why. 94. Fid all umbers a ad b so that 0, a, b, ab are the first 4 terms of a arithmetic sequece.
9. Evaluate the sum 000 990. 96. If 4 4 7, the what is the value of 8 8? {Hit: 4 4 7 8 7 ad {Hit: a b a ba b.} 4 4 7 8 7. So add the two equatios together to get 8 8 7 8 8 6 of 97. Assumig that log4 6 usig A, B, or C. 98. Assumig that log4 9 usig A, B, or C. 99. Assumig that log 7 usig A, B, or C. 00. Assumig that log 9 usig A, B, or C.. If you kew the value, the you d have the aswer. Well, what s A, log40 A, log4 A, log A, log B, ad log 4 7 B, ad log 4 6 B, ad log 9 B, ad log 7 0. Let s prove that log is a irratioal umber. Suppose that log?} C, the epress the value of log4 {Hit: 7 4.} 8 C, the epress the value of log4 {Hit: 8 9.} C, the epress the value of log 4 {Hit: 4 7.} 9 C, the epress the value of log {Hit: 7.} m for positive itegers m m m ad. The 0 ad therefore 0. Compare the oes digits of powers of 0 to the oes digits of powers of to arrive at a cotradictio.
m 0. Let s prove that log is a irratioal umber. Suppose that log for positive m m itegers m ad. The ad therefore. Compare the oes digits of powers of to the oes digits of powers of to arrive at a cotradictio. m 0. Let s prove that log is a irratioal umber. Suppose that log for positive m m itegers m ad. The ad therefore. Compare the oes digits of powers of to the oes digits of powers of to arrive at a cotradictio. 04. a) Show that if bd, 0, the a b c is equivalet to ad bc. d {Hit: If a b c d the a c 0 ad bc b d bd 0 ad bc, ad ad bc ad bc 0 0.} bd b) Show that if bd, 0 ad a b c the a a c c. d b b d d a a a c) Use Mathematical Iductio to show that if b, b,, b 0 ad, the b b b a a a a a b b b b b. Use the - fuctios f ad g which are defied by the followig tables to fid the followig. f g - - 0 0 4 4-0 - 0. f g 4 06. g f 0 07. f g 08. f g 09. g f 0. Solve g 0.
. Solve f g 0.. Solve f g 4. Solve g f. 0.. Solve f g.. For the oe-to-oe fuctios f 0,,,,, ad g,0,,, 4,, f g,,, ad g f 0,0,,, so they could be described as f g ad g f. Are f ad g iverses? Eplai. 6. You might thik that if f g for all i the domai of g ad g f is defied for all i the domai of f, that g f. For g,,,4 ad f,, 4,, 6,, f g for all i the domai of g, g f is defied for all i the domai f. Is it the case that g f for all i the domai of f? Eplai. 7. A fuctio is called a ivolutio if it is its ow iverse. Which of the followig fuctios are ivolutios? What type of symmetry does the graph of a ivolutio have? a) f b) f c) f ; 0 d) f e) f f) f ; 0 ;0 g) f ; 0 0 ; 0 ;0