Algebra 8 Exercise (2) PDF

28. Write each of the following statements in the form [*a*^*x*^ = *y*]{.math.inline} 1. a) \ [log~*m*~*c* = *d*]{.math.display}\ 2. b) \ [log~*b*~*p* = *q*]{.math.display}\ 3. c) \ [log *s* = *t*]{.math.display}\ 4. d) \ [ln *z* = *r*]{.math.display}\ 5. Find the value of...

28. Write each of the following statements in the form [*a*^*x*^ = *y*]{.math.inline} 1. a) \ [log~*m*~*c* = *d*]{.math.display}\ 2. b) \ [log~*b*~*p* = *q*]{.math.display}\ 3. c) \ [log *s* = *t*]{.math.display}\ 4. d) \ [ln *z* = *r*]{.math.display}\ 5. Find the value of [*x*]{.math.inline} in each of the following 6. a) \ [log~2~8 = *x*]{.math.display}\ 7. b) \ [log~3~81 = *x*]{.math.display}\ 8. c) \ [log~*x*~216 = 3]{.math.display}\ 9. d) \ [log~*x*~16 = 0.5]{.math.display}\ 10. e) \ [log *x* =  − 2]{.math.display}\ 11. f) \ [log~*a*~1]{.math.display}\ 12. g) \ [log~*a*~*a*]{.math.display}\ 13. h) \ [log~8~2]{.math.display}\ 14. i) \ [\$\$\\log\_{27}\\frac{1}{9}\$\$]{.math.display}\ 15. Evaluate the following 16. a) \ [log~2~32]{.math.display}\ b) \ [log~6~36]{.math.display}\ c) \ [log~3~625]{.math.display}\ d) \ [\$\$\\log\_{2}\\frac{1}{2}\$\$]{.math.display}\ 17. e) \ [\$\$\\log\_{2}\\frac{1}{4}\$\$]{.math.display}\ f) \ [log~*a*~1]{.math.display}\ g) \ [log~*a*~*a*]{.math.display}\ h) \ [log~8~2]{.math.display}\ 18. i) \ [\$\$\\log\_{8}\\frac{1}{2}\$\$]{.math.display}\ j) \ [log~27~3]{.math.display}\ k) \ [log~8~4]{.math.display}\ l) \ [\$\$\\log\_{27}\\frac{1}{9}\$\$]{.math.display}\ 19. Express each of the following in terms of [log *a*, log *b*]{.math.inline} and [log *c*.]{.math.inline} 20. a) \ [log (abc)]{.math.display}\ b) \ [log (*a*^2^bc)]{.math.display}\ c) \ [log (*a*^3^*b*^2^*c*)]{.math.display}\ d) \ [\$\$\\log\\left( a\\sqrt{b} \\right)\$\$]{.math.display}\ 21. e) \ [\$\$\\log\\left( \\frac{\\text{ab}}{c} \\right)\$\$]{.math.display}\ f) \ [\$\$\\log\\left( \\frac{a\^{2}}{bc\^{3}} \\right)\$\$]{.math.display}\ g) \ [\$\$\\log\\left( \\frac{1}{\\text{ab}} \\right)\$\$]{.math.display}\ h) \ [\$\$\\log\\left( a\\sqrt{\\frac{b}{c\^{3}}} \\right)\$\$]{.math.display}\ 22. i) \ [log (10*ab*^2^)]{.math.display}\ j) \ [\$\$\\log\\left( \\frac{\\sqrt{10a}}{b\^{2}} \\right)\$\$]{.math.display}\ 23. Express as a single logarithm 24. a) \ [2log~*a*~5 + log~*a*~4 − log~*a*~10]{.math.display}\ 25. b) \ [2log~*a*~3 − log~*a*~15 + log~*a*~5]{.math.display}\ 26. c) \ [\$\$\\log\_{a}12 - \\left( \\frac{1}{2}\\log\_{a}9 + \\frac{1}{3}\\log\_{a}8 \\right)\$\$]{.math.display}\ 27. d) \ [\$\$2\\log\_{a}\\frac{1}{2} + 3\\log\_{a}4 - \\log\_{a}\\frac{4}{3}\$\$]{.math.display}\ +-----------------------+-----------------------+-----------------------+ | 67. | Express each of the | | | | following in the form | | | | [log~*a*~*f*(*x*)]{.m | | | | ath | | | |.inline} | | +=======================+=======================+=======================+ | 29. | a) | \ | | | | [2log~*a*~*x*]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 30. | b) | \ | | | | [log~*a*~*x* + log~*a | | | | *~(*x* + 3)]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 31. | c) | \ | | | | [log~*a*~(*x* + 1) −  | | | | log~*a*~2]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 32. | d) | \ | | | | [log~*a*~(*x*^2^−1) − | | | |  log~*a*~(*x*+1)]{.ma | | | | th | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 33. | e) | \ | | | | [2log~*a*~*x* − log~* | | | | a*~*x*(*x* + 1)]{.mat | | | | h | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 34. | f) | \ | | | | [3log~*a*~*x* + log~* | | | | a*~(*x*+1)]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 35. | g) | \ | | | | [4log~*a*~*x* − log~* | | | | a*~(*x*^2^+*x*^3^)]{. | | | | math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 36. | h) | \ | | | | [\$\$\\frac{5}{2}\\lo | | | | g\_{a}x | | | | + \\log\_{a}\\left( x | | | | + 1 \\right) - | | | | \\log\_{a}\\sqrt{x}\$ | | | | \$]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 37. | Find the value of | | | | [*x*]{.math.inline} | | | | in each of the | | | | following | | +-----------------------+-----------------------+-----------------------+ | 38. | a) | \ | | | | [3log~2~4 = *x*]{.mat | | | | h | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 39. | b) | \ | | | | [2log~4~8 = *x*]{.mat | | | | h | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 40. | c) | \ | | | | [4log~*x*~2 = 0.5]{.m | | | | ath | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 41. | Use your calculator | | | | to find [*x*]{.math | | | |.inline} correct to 2 | | | | decimal places. | | +-----------------------+-----------------------+-----------------------+ | 42. | a) | \ | | | | [log 23 = *x*]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 43. | b) | \ | | | | [ln 23 = *x*]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 44. | c) | \ | | | | [log *x* = 0.32]{.mat | | | | h | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 45. | d) | \ | | | | [log *x* = 1.4]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 46. | e) | \ | | | | [log *x* =  − 0.65]{. | | | | math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 47. | f) | \ | | | | [ln *x* = 3.1 ]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 48. | g) | \ | | | | [ln *x* = 0.13]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 49. | h) | \ | | | | [ln *x* =  − 0.45]{.m | | | | ath | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 50. | Given that | | | | [log 2 = 0.3]{.math | | | |.inline} show that | | | | [log 5 = 0.7]{.math | | | |.inline}. Using these | | | | values, find the | | | | values of the | | | | following | | +-----------------------+-----------------------+-----------------------+ | 51. | a) | \ | | | | [log 2.5]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 52. | b) | \ | | | | [log 4]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 53. | c) | \ | | | | [log 25]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 54. | d) | \ | | | | [log 32]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 55. | e) | \ | | | | [log 0.5]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 56. | f) | \ | | | | [log~2~5]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 57. | g) | \ | | | | [log~5~10]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 58. | Given that | | | | [ln 5 = 1.6]{.math | | | |.inline} and | | | | [ln 11 = 2.4]{.math | | | |.inline} find the | | | | values of | | +-----------------------+-----------------------+-----------------------+ | 59. | a) | \ | | | | [ln 625]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 60. | b) | \ | | | | [ln 55]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 61. | c) | \ | | | | [ln 2.2]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 62. | d) | \ | | | | [ln 275]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 63. | e) | \ | | | | [ln 0.2]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 64. | f) | \ | | | | [ln 0.04]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 65. | g) | \ | | | | [log~5~11]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 66. | Plot the graphs of | | | | [*y* = 2^*x*^]{.math | | | |.inline} and | | | | [*y* = log~2~*x*]{.ma | | | | th | | | |.inline} on the same | | | | set of axes. | | +-----------------------+-----------------------+-----------------------+ +-----------------------+-----------------------+-----------------------+ | 88. | Show that | | | | [log~*a*~*b* × log~*b | | | | *~*a* = 1]{.math | | | |.inline} and hence | | | | evaluate | | +=======================+=======================+=======================+ | 68. | a) | \ | | | | [log~3~8 × log~2~9]{. | | | | math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 69. | b) | \ | | | | [log~5~64 × 2log~4~25 | | | | ]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 70. | c) | \ | | | | [log~10~8 × log~2~100 | | | | 0]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 71. | Given that | | | | [log 2 = 0.3, log 3 = | | | |  0.48]{.math | | | |.inline} and | | | | [log 17 = 1.23]{.math | | | |.inline} evaluate | | +-----------------------+-----------------------+-----------------------+ | 72. | a) | \ | | | | [log~2~3]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 73. | b) | \ | | | | [log~2~17]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 74. | Show graphically that | | | | the following figures | | | | support the belief | | | | that a relationship | | | | of the form | | | | [*y* = *ak*^*x*^]{.ma | | | | th | | | |.inline} exists and | | | | use your graph to | | | | find approximate | | | | values for the | | | | constants [*a*]{.math | | | |.inline} and | | | | [*k*]{.math.inline}. | | | | | | | | ------------------- | | | | ----- ----- ----- --- | | | | -- ----- ------ ----- | | | | - ------ | | | | \ | | | | 1 2 3 | | | | 4 6 8 | | | | 10 | | | | [*x*]{.math.displa | | | | y}\ | | | | | | | | | | | | | | | | \ | | | | 3.5 4.9 6.9 | | | | 9.6 18.8 36.9 | | | | 72.3 | | | | [*y*]{.math.displa | | | | y}\ | | | | | | | | | | | | ------------------- | | | | ----- ----- ----- --- | | | | -- ----- ------ ----- | | | | - ------ | | +-----------------------+-----------------------+-----------------------+ | 75. | Show graphically that | | | | the following figures | | | | support the belief | | | | that a relationship | | | | of the form | | | | [*y* = *ak*^*x*^2^^]{ | | | |.math | | | |.inline} exists and | | | | use your graph to | | | | find approximate | | | | values for the | | | | constants [*a*]{.math | | | |.inline} and | | | | [*k*]{.math.inline}. | | | | | | | | ------------------- | | | | --------------------- | | | | --------------------- | | | | --------------------- | | | | --------------------- | | | | ------------------- | | | | \ | | | | 0.5 1 1. | | | | 5 2 2.5 | | | | | | | | 3 | | | | [*x*]{.math.displa | | | | y}\ | | | | | | | | | | | | | | | | ------------------- | | | | ----- ----- ------ -- | | | | ----- ------ -------- | | | | --------------------- | | | | ----- --------------- | | | | ------------------- | | | | \ | | | | 3.6 13.8 12 | | | | 9.6 2981 \ | | | | | | | | \ | | | | [*y*]{.math.displa | | | | y}\ | | | | [1.679 × | | | |  10^5^]{.math.displa | | | | y}\ [2.318 × 10^7^] | | | | {.math.display}\ | | | | | | | | ------------------- | | | | --------------------- | | | | --------------------- | | | | --------------------- | | | | --------------------- | | | | ------------------- | | +-----------------------+-----------------------+-----------------------+ | 76. | Show graphically that | | | | the following figures | | | | support the belief | | | | that a relationship | | | | of the form | | | | [*y* = *ax*^*k*^]{.ma | | | | th | | | |.inline} exists and | | | | use your graph to | | | | find approximate | | | | values for the | | | | constants [*a*]{.math | | | |.inline} and | | | | [*k*]{.math.inline}. | | | | | | | | ------------------- | | | | --------------------- | | | | --------------------- | | | | -------- | | | | \ | | | | 5 8 1 | | | | 2 14 20 | | | | 30 | | | | [*x*]{.math.displa | | | | y}\ | | | | | | | | | | | | ------------------- | | | | ----- ------ ------ - | | | | ----- ------- ------- | | | | ------- | | | | \ | | | | 22.4 45.3 8 | | | | 3.1 104.8 178.9 | | | | 328.6 | | | | [*y*]{.math.displa | | | | y}\ | | | | | | | | | | | | | | | | ------------------- | | | | --------------------- | | | | --------------------- | | | | -------- | | +-----------------------+-----------------------+-----------------------+ | 77. | Show graphically that | | | | the following figures | | | | support the belief | | | | that a relationship | | | | of the form | | | | [*y* = *a*(2+*x*)^*k* | | | | ^]{.math | | | |.inline} exists and | | | | use your graph to | | | | find approximate | | | | values for the | | | | constants [*a*]{.math | | | |.inline} and | | | | [*k*]{.math.inline}. | | | | | | | | ------------------- | | | | --------------------- | | | | --------------------- | | | | --------- | | | | \ | | | | 1 3 8 | | | | 10 15 20 | | | | 24 | | | | [*x*]{.math.displa | | | | y}\ | | | | | | | | | | | | ------------------- | | | | ----- ----- ----- --- | | | | -- ------ ------ ---- | | | | -- ------ | | | | \ | | | | 5.2 6.7 9.5 | | | | 10.4 12.4 14.1 | | | | 15.3 | | | | [*y*]{.math.displa | | | | y}\ | | | | | | | | | | | | | | | | ------------------- | | | | --------------------- | | | | --------------------- | | | | --------- | | +-----------------------+-----------------------+-----------------------+ | 78. | Using the [ln ]{.math | | | |.inline} or | | | | [log ]{.math.inline} | | | | function on a | | | | calculator solve the | | | | following equations | | | | giving your answers | | | | correct to 3 | | | | significant figures. | | +-----------------------+-----------------------+-----------------------+ | 79. | a) | \ | | | | [2^*x*^ = 5]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 80. | b) | \ | | | | [3^*x*^ = 17]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 81. | c) | \ | | | | [5^*x*^ = 10]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 82. | d) | \ | | | | [4^*x*^ = 12]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 83. | e) | \ | | | | [3^*x*^ = 365]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 84. | f) | \ | | | | [3^*x* + 1^ = 2^*x*^] | | | | {.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 85. | g) | \ | | | | [4^*x* + 1^ = 3^2 − * | | | | x*^]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 86. | h) | \ | | | | [5^*x* − 1^ = 2^3*x*  | | | | − 1^]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 87. | i) | \ | | | | [6^3*x* + 1^ = 7^2 −  | | | | *x*^]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ +-----------------------+-----------------------+-----------------------+ | 111. | Solve the following | | | | equations giving your | | | | answers correct to 3 | | | | significant figures. | | +=======================+=======================+=======================+ | 89. | a) | \ | | | | [2^2*x*^ − 3 × 2^*x*^ | | | |  − 4 = 0]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 90. | b) | \ | | | | [4^2*x*^ + 4^*x* + 2^ | | | |  − 80 = 0]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 91. | c) | \ | | | | [3^2*x* + 1^ + 3 = 10 | | | |  × 3^*x*^]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 92. | d) | \ | | | | [4^2*x*^ − 4^*x* + 2^ | | | |  = 80]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 93. | e) | \ | | | | [2^2*x* + 1^ − 7 × 2^ | | | | *x*^ + 6 = 0]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 94. | f) | \ | | | | [\$\$1 + \\log\_{2}x | | | | = | | | | \\frac{12}{\\log\_{2} | | | | x}\$\$]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 95. | Solve the following | | | | logarithmic equations | | +-----------------------+-----------------------+-----------------------+ | 96. | a) | \ | | | | [log~*a*~*x* + log~*a | | | | *~4 − log~*a*~5 = log | | | | ~*a*~12]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 97. | b) | \ | | | | [\$\$\\log\_{4}x - | | | | \\log\_{4}7 = | | | | \\frac{3}{2}\$\$]{.ma | | | | th | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 98. | c) | \ | | | | [log~9~*x* + log~9~*x | | | | *^2^ + log~9~*x*^3^ + | | | |  log~9~*x*^4^ = 5]{.m | | | | ath | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 99. | d) | \ | | | | [log~9~4 + log~3~*x*  | | | | = 3]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 100. | e) | \ | | | | [log~2~*x* − log~8~*x | | | | * = 4]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 101. | f) | \ | | | | [log~5~*x* + 2log~*x* | | | | ~5 = 3]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 102. | Solve the following | | | | simultaneous | | | | equations | | +-----------------------+-----------------------+-----------------------+ | 103. | a) | \ | | | | [2log~*x*~*y* = 1]{.m | | | | ath | | | |.display}\ | | | | | | | | \ | | | | [*xy* = 64]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 104. | b) | \ | | | | [6log~3~*x* + 6log~27 | | | | ~*y* = 7]{.math | | | |.display}\ | | | | | | | | \ | | | | [4log~9~*x* + 4log~3~ | | | | *y* = 9]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 105. | c) | \ | | | | [*y*log~2~8 = *x*]{.m | | | | ath | | | |.display}\ | | | | | | | | \ | | | | [2^*x*^ + 8^*y*^ = 81 | | | | 92]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 106. | d) | \ | | | | [log *x* − log 2 = 2l | | | | og *y*]{.math | | | |.display}\ | | | | | | | | \ | | | | [*x* − 5*y* + 2 = 0]{ | | | |.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 107. | e) | \ | | | | [5^*x* + 2^ + 7^*x* + | | | |  1^ = 3468]{.math | | | |.display}\ | | | | | | | | \ | | | | [7^*x*^ = 5^*x*^ − 76 | | | | ]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 108. | f) | \ | | | | [log~2~*x* + 2log~4~* | | | | y*]{.math | | | |.display}\ | | | | | | | | \ | | | | [*x* + 12*y* = 52]{.m | | | | ath | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 109. | g) | \ | | | | [ln 6 + ln (*x*−3) =  | | | | 2ln *y*]{.math | | | |.display}\ | | | | | | | | \ | | | | [2*y* − *x* = 3]{.mat | | | | h | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ | 110. | h) | \ | | | | [2 + log~2~(2*x*+1) = | | | |  2log~2~*y*]{.math | | | |.display}\ | | | | | | | | \ | | | | [*x* = 22 − *y*]{.mat | | | | h | | | |.display}\ | +-----------------------+-----------------------+-----------------------+