Math 141 – TEST # 4 Sections 4



Math 141 – TEST # 4 Sections 4.3 – 4.8 Fall 2008

|1. |A. |ex = 36 |LN means “base 3”, so log base e of the product 36 = exponent x |

| |B. |517 = x |Log base 5 of product x = exponent 17 |

|2. |A. |Ln c = y |Log base e of product c = exponent y |

| |B. |Log 64 [pic]= [pic] |Log base 64 of product 1/8 = exponent -1/2 |

|3. |A. |5 |Let log 3 243 = x. Change to exponential form: 3x = 243. Prime factor 243: 243 = 35, so 3x = 35. Same base ( |

| | | |exponents are =, so x = 5. |

| |B. |4 |Let ln (e4) = x. Change to expo form: ex = e4. Same base ( exponents =, so x = 4. |

| |C. |[pic] |Let log 8 4 = x. Change to expo form: 8x = 4. Prime factor both 8 and 4 ( (23)x = 22. Power of power ( multiply |

| | | |exponents ( 23x = 22. Same base ( exponents = ( 3x = 2. x = 2/3. |

| |D. |[pic] |Let log 2 [pic]= x. Change everything to expo form, including the radical (fraction expo). 2x = 1281/2 ( Prime |

| | | |factor 128 ( 2x = (2 ½ )7 ( Multiply exponents when power of power ( 2x = 27/2 ( Same base ( Expo = ( x = 7/2. |

|4. |A. |[pic] |Write the radical as an exponent (expo / root) ( Log 2 (ab)1/3 or |

| | | |log 2 (a1/3 b1/3 ). Use Rule 4, log 2 a1/3 + log 2 b1/3 ( Use Rule 6, [pic] |

| |B. |2(log 6 X – log 6 3 - log 6 |Use Rule 5, log 6 X2 – log 6 (3Y). Use Rule 6, 2(log 6 X – log 6 (3Y). Use Rule 4, 2(log 6 X – (log 6 3 + log 6 |

| | |Y |Y). Distributing gives answer. |

| |C. |log 3 U + 2( log 3 V – log 3 |Use Rule 5 ( log 3 (UV2) – log 3 Z. Use Rule 4 ( log 3 U + log 3 V2 – log 3 Z. Use Rule 6 ( log 3 U + 2( log 3 V|

| | |Z |– log 3 Z. |

| |D. |8 ( log 2 a + |Use Rule 6 ( 4 ( log 2 ([pic]). Use Rule 4 ( 4 ( [log 2 (a2) + log 2 (b1/2)]( Use Rule 6, then distributive |

| | |2 ( log 2 b |property ( 4 ( [2(log 2 a + (1/2)(log 2 b) ( 8 ( log 2 a + 2 ( log 2 b |

|5. |A. |[pic] |Use Rule 6 ( log 3 X – (log 3 Y2 + log 3 Z1/3). Use Rule 4 and fraction exponents mean radicals ( log 3 X – (log|

| | | |3 Y2( [pic]). Use Rule 5 to finish. |

| |B. |ln [pic] |Use Rule 6 ( ln (x + y)1/2 – ln x3. Use Rule 5 and fraction exponents mean radicals to finish. |

|6. |A. | |Write in expo form ( 21/2 = [pic]. Square both sides ( (21/2)2 = [pic] ( 2 = x + 4 ( x = -2. CHECK required |

| | |- 2 |because you cannot take the log of a negative number, but log 2 (-2 + 4) is the log of a positive number, so the|

| | | |answer is okay. |

|6. |B. |4 |Expo form, try same base by prime factorization ( (32)2-x = 1 / 34 ( 34-2x = 3-4. Same base ( equal expo ( 4 – |

| | | |2x = - 4 ( - 2x = - 8 ( x = 4 |

| |C. |[pic] |Prime factor the bases ( (24)3x-1 = (2 -2)x. Power of power mean to multiply exponents ( 2 12x – 4 = 2 -2x Same |

| | | |base ( equal exponents ( 12x – 4 = - 2x ( 14x = 4 ( x = 2 / 7 |

| |D. |[pic] |Same bases won’t work. X is in both exponents so changing to log form won’t work well, therefore, take the LOG |

| | | |of BOTH sides of the equation. |

| | | |Log (8 2x) = Log (3 x+1) ( Rule 6: 2x (log 8) = (x + 1)(log 3) ( Distributive property: 2x (log 8) = x (log 3) +|

| | | |log 3 ( Get x’s on same side: 2x (log 8) – x (log 3) = log 3 ( Factor out x: x ( 2 log 8 – log 3) = log 3 ( |

| | | |Divide by ( 2 log 8 – log 3) ( Use calculator: 0.3589918548 |

| |E. |NO solution |Put together as a single log: Rule 6 ( log 6 (x – 1) – log 6 x2 = 1 ( [pic]. Write in exponential form: [pic]. |

| | | |Multiply both sides by x2 ( x – 1 = 6x2. Quadratic equation = 0 and factor or use quadratic formula ( 6x2 – x + |

| | | |1 = 0 ( [pic] ( complex, not real, answers. |

| |F. |[pic]-0.695 |Same bases not possible ( take LOG of BOTH sides ( Log (4 x-2) = |

| | | |Log (63x). Rule 6: (x – 2)log 4 = 3x (log 6). Distributive property: x (log 4) – 2(log 4) = 3x (log 6) ( Get x’s|

| | | |on same side of = ( x (log 4) – 3x (log 6) = 2 (log 4) ( Factor out x: x (log 4 – 3 log 6) = 2 log 4 ( Divide by|

| | | |(log 4 – 3 log 6) ( [pic] Use calculator: -0.6950613715. Negative exponents are acceptable. |

| |G. |1 |Like E above: [pic]. Write in exponential form: [pic] Multiply by x2: x + 2 = 3x2 ( Quadratic equation = 0 and |

| | | |factor ( 3x2 – x – 2 = 0 ( (3x + 2)(x – 1) = 0 ( 3x + 2 = 0 or x – 1 = 0 ( x = -2 / 3, 1 CHECK required for log|

| | | |equations: Throw out – 2/3 because you can’t have log(negative number). |

| |H. |[pic]4.301 |Can’t change to same base, so take LOG of BOTH sides ( Log (5x) = Log (3x + 2) ( Use Rule 6: x (log 5) = (x + 2)|

| | | |log 3 ( Distributive Property: x (log 5) = x (log 3) + 2 (log 3) ( Get x’s on same side of = ( x (log 5) – x |

| | | |(log 3) = 2 (log 3) ( Factor out x: x (log 5 – log 3) = 2 (log 3) ( Divide by (log 5 – log 3) and use |

| | | |calculator. X = 4.301320206 |

| |I. |3, 4 |Rule 6: log (7x – 12) = log x2. Log with same base, so products are = ( 7x – 12 = x2 ( Quadratic eq = 0 and |

| | | |factor ( x2 – 7x + 12 = 0 ( (x – 3) (x – 4) = 0 ( x = 3, 4 |

| |J. |[pic] - 0.609 |Exponential form ( change to log form ( ln 5 = 1 – x ( x = 1 – ln 5. Use calculator x = -0.6094379124 |

| |K. |- 79 |Log form ( change to expo form ( 2 – x = 34 ( 2 – x = 81 ( - x = 79 ( x = -79 |

| |L. |[pic] |Same base by prime factorization and fractions create negative exponents ( (34)2x = 3 -1 ( Power of power means |

| | | |to multiply exponents ( 3 8x = 3 -1 ( Same base ( = expo ( 8x = - 1 |

|6. |M. |[pic] |Same base and rewrite radical in exponent form: 21/2 = (23)x. Multiply exponents ( 21/2 = 23x. ½ = 3x ( x = |

| | | |1/6 |

| |N. |3 |X is isolated, so the problem is solved. Use change-of-base formula to find the numerical value for x. [pic]3|

| | | | |

| | | |(Remember that the log of the Base goes on Bottom.) |

| |O. |3 |Change to expo form ( x4 = 81 ( Take the 4th root of each side ( [pic] ( x = 3 (because 81 = 34) |

| |P. |2 |Change to expo form ( 41/2 = x = [pic]= 2 |

| |Q. |[pic] |Same base ( (24)3x = (23)x+1 ( 12x = 3x + 3 ( 9x = 3 ( x = 1/3 |

| |R. |10,000 |Log with no base is the common log with understood base 10. Change to expo form ( 104 = x = 10000 |

|7. |A. |[pic] 1.292 |“Evaluate” means numerical answer. Use change-of-base formula ([pic]= 1.292029674 |

| |B. |[pic] 2.262 |Change of base formula: [pic]= 2.261859507 |

|8. |1st ? |[pic] 11.5 yrs |[pic]TVM Solver: N= ?, I% = 8, PV = -2000, Pmt = 0, FV = 5000, P/Y = C/Y = 12 ( N = 137.9 months = 12t ( t = |

| | | |11.5 yrs |

| |2nd ? |[pic] 11.5 yrs |Compounded continuously ( End = Beg ( erate X time ( 5000 = 2000e.08t (2.5 = e.08t ( ln 2.5 = .08t ( t = (ln |

| | | |2.5) / .08 = 11.45363415 |

|9. | |$ 2433.98 or |[pic]( N = 24, I% = 3.5, PV = ?, Pmt = 0, FV = 3000, P/Y = C/Y = 4 ( PV = - 2433.974978 |

| | |$ 2434 | |

|10. | |$ 2992.73 |[pic]( N=52 x 3 = 156, I% = 6, PV = - 2500, Pmt = 0, FV = ?, P/Y = C/Y = 52. FV = 2992.732847 |

|11. | |5832 mosquitoes after 3 days |Use the first case to find the rate of growth of the mosquitoes ( |

| | | |1800 = 1000 er(1 day ( 1.8 = er ( ln 1.8 = r = .5877866649 or 58.8% |

| | | |End = 1000 e.588 x 3 ( 5832 |

| | |3.9 or 4 days to reach 10,000 |10000 = 1000 e.588 t ( 10 = e.588 t ( ln 10 = .588 t ( t = (ln 10) / .588 ( t = 3.917382327 |

| | |mosquitoes |NOTE: I stored ln 1.8 as X in my calculator and did not use the rounded version. If you used the rounded |

| | | |version of .588, your answer will be slightly different from the key. |

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