MTH 112 Practice Test 3 Sections 3.3, 3., 3., 1.9, 7., 7., 8.1, 8.2 Use properties of logarithms to epand the logarithmic epression as much as possible. Where possible, evaluate logarithmic epressions without using a calculator. 1) log - 6 2) log 2 2 7w Solve the logarithmic equation. Be sure to reject an value that is not in the domain of the original logarithmic epressions. Give the eact answer. 13) log 3 ( - 1) = -1 1) log ( + 3) + log ( - 3) = 2 1) ln 6 + ln ( - 1) = 0 3) log 7 Use properties of logarithms to condense the logarithmic epression. Write the epression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic epressions. ) 3ln a - 9 ln b - ln c ) log3 2 + 1 log 3 (r - 2) - 1 2 log 3 r Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places 6) log π 17 7) log 26 382 Solve the equation b epressing each side as a power of the same base and then equating eponents. 8) (1 + 2) = 6 9) 2(7-3) = 1 Solve the eponential equation. Epress the solution set in terms of natural logarithms. ) + 7 = 3 11) + = 2 + 16) log 3 ( + 6) + log 3 ( - 6) - log 3 = 2 17) log 6 ( + 6) = 3 - log 6 18) log ( + 20) - log 2 = log (3 + ) 19) ln ( - 6) + ln ( + 1) = ln ( - 1) Solve the problem. 20) Find out how long it takes a $2600 investment to double if it is invested at 8% compounded semiannuall. Round to the nearest tenth of a ear. Use the formula A = P 1 + r nt. n Solve. 21) The value of a particular investment follows a pattern of eponential growth. In the ear 2000, ou invested mone in a mone market account. The value of our investment t ears after 2000 is given b the eponential growth model A = 30e0.06t. When will the account be worth $3902? 22) The function A = A0e-0.00693 models the amount in pounds of a particular radioactive material stored in a concrete vault, where is the number of ears since the material was put into the vault. If 900 pounds of the material are initiall put into the vault, how man pounds will be left after 30 ears? 12) e + = 2 1
23) The population of a particular countr was 22 million in 198; in 199, it was 31 million. The eponential growth function A =22ekt describes the population of this countr t ears after 198. Use the fact that ears after 198 the population increased b 9 million to find k to three decimal places. Graph the equation. 33) (- 6)2 + ( - 3)2 = 9 2) The function A = A0e-0.0077 models the amount in pounds of a particular radioactive material stored in a concrete vault, where is the number of ears since the material was put into the vault. If 700 pounds of the material are placed in the vault, how much time will need to pass for onl pounds to remain? - - 2) The half-life of silicon-32 is 7 ears. If 0 grams is present now, how much will be present in 200 ears? (Round our answer to three decimal places.) Graph the equation and state its domain and range. Use interval notation 3) 2 + 2 = 9 26) The population of a certain countr is growing at a rate of 1.8% per ear. How long will it take for this countr's population to double? Use the formula t = ln 2, which gives the time, t, k for a population with growth rate k, to double. (Round to the nearest whole ear.) - - Write the standard form of the equation of the circle with the given center and radius. 27) (, 0); 3 28) (8, -3); 9 Solve the sstem b the substitution method. 3) 8 - = 1 = 2 + 6 Find the center and the radius of the circle. 29) ( + 8)2 + ( - 1)2 = Complete the square and write the equation in standard form. Then give the center and radius of the circle. 30) 2 + 2 - - + 29 = 36 31) 2 + 2 = 0 32) 2 + 2 - - 8 + 29 = 0 36) 2 + 2 = 113 + = 1 37) = 6 + = -1 Solve the sstem b the addition method. 38) 2-2 = -3 22 + 22 = 0 Solve the sstem b the substitution method. 39) = 2-3 2 + 2 = 2
Graph the inequalit. 0) (- 1)2 + ( - )2 > 9 3) > 2 + 6 60 - - - - 1) 2 - ) 2 + 2 36 2 + 2 2 - - - - Graph the solution set of the sstem of inequalities or indicate that the sstem has no solution. 2) + 2 2-0 ) 2 + 2 36-8 + 3-2 - - - - 3
6) 2 + 2 9-2 > 0 Solve the sstem of equations using matrices. Use Gaussian elimination with back-substitution. 2) 6 - - 6z = -3-6 - z = -38 9 + z = 17-3) - + z = 11 2 + z = 3 + 2 + z = 11 - Write the augmented matri for the sstem of equations. 7) + z = 0 + z = 9 + 8 + 7z = 8 Write the sstem of linear equations represented b the augmented matri. Use,, z, and, if necessar, w for the variables. 8) 9 8-2 0 7 0 2 Write the sstem of linear equations represented b the augmented matri. Use,, z, and, if necessar, w for the variables. Then use back-substitution to find the solution. 9) 1 2 9-8 0 1-3 0 0 1-6 Solve the sstem of equations using matrices. Use Gauss-Jordan elimination. ) 3 - - 7z = 7 6 + - 3z = 67-6 - 3 + z = -62 Use Gaussian elimination to find the complete solution to the sstem of equations, or state that none eists. ) + + z = 7 - + 2z = 7 2 + 3z = 1 6) + 8 + 8z = 8 7 + 7 + z = 1 8 + 1 + 9z = -9 7) - + 3z = 12 + + 6z =-32 + 3 + 9z = 20 8) + + z = 9 2-3 + z = 7 - + 3z = -2 Perform the matri row operation (or operations) and write the new matri. 0) - 1 1-0 -2 -R1 + R2-1 3-3 -1 1) 33 12-3 -1 1 13-3 0 2-7 21 1 3 R 1
Answer Ke Testname: MTH 112 PRACTICETEST3 1) log ( - 6) - log 2) 2log 2-7log 2 - log 2 w 33) 3) 1 2 log 7 + 1 2 log ) ln a 3 b9 c - ) log3 32 r - 2 r 6) 2.70 7) 1.828 8) {1} 9) {3} ln 3 ) ln - 7 11) ln - ln ln - 2 ln 12) {ln 2 - } 13) 3 1) {} 1) { 7 6 } 16) {12} 17) {12} 12 18) 19) 20) 8.8 ears 21) 200 22) 731 pounds 23) 0.03 2) 200 ears 2) 1.131 26) 39 ears 27) ( + )2 + 2 = 9 28) ( - 8)2 + ( + 3)2 = 81 29) (-8, 1), r = 2 30) ( - 2)2 + ( - )2 = 36 (2, ), r = 6 31) 2 + 2 = (0, 0), r = 32) ( - )2 +( - )2 = 12 (, ), r = 2 3 - Domain = (3, 9), Range = (0, 6) 3) - - Domain = (-7, 7); Range = (-7, 7) 3) {(7, ), (1, 7)} 36) {(8, 7), (7, 8)} 37) {(-7, -8), (-8, -7)} 38) {(3, ), (-3, ), (3, -), (-3, -)} 39) {(-2, 1), (-1, -2), (1, -2), (2, 1)} 0) - -
Answer Ke Testname: MTH 112 PRACTICETEST3 1) ) 6 2 - - -8-6 - -2 2 6 8-2 - -6 2) ) 6 2-6 - -2 2 6 - -2 - - 3) -6 - - - - 6) 7) 0 0 0 9 8 7 8 8) + 9 + 8z = -2 + z = 7 + = 2 9) {(7,-1,-6)} 0) - 1 1-20 20 1-6 -1 3-3 -1 6
Answer Ke Testname: MTH 112 PRACTICETEST3 1) 11-1 - 1 13-3 0 2-7 21 2) {(1, 1, 8)} 3) {(0,, 3)} ) {(7, 7, 1)} ) {(- 3z 2 + 7, z 2, z)} 6) 7) 8) {(- 7z + 3, 2z + 11, z)} 7