Math 141 – TEST # 4 Sections 4.3 – 4.8Fall 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= / 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. / / 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. / / Let log 2= 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. / / 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,
B. / 2log 6 X – log 6 3 - log 6 Y / 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). Distributing gives answer.
C. / log 3 U + 2 log 3 V – log 3 Z / 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 – log 3 Z.
D. / 8  log 2 a +
2  log 2 b / Use Rule 6  4  log 2 (). Use Rule 4  4  [log 2 (a2) + log 2 (b1/2)] Use Rule 6, then distributive property  4  [2log 2 a + (1/2)(log 2 b)  8  log 2 a + 2  log 2 b
5. / A. / / 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). Use Rule 5 to finish.
B. / ln / Use Rule 6  ln (x + y)1/2 – ln x3. Use Rule 5 and fraction exponents mean radicals to finish.
6. / A. / - 2 / Write in expo form  21/2 = . Square both sides  (21/2)2 =  2 = x + 4  x = -2. CHECK required 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. / / 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. / / 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 . Write in exponential form: . Multiply both sides by x2 x – 1 = 6x2. Quadratic equation = 0 and factor or use quadratic formula  6x2 – x + 1 = 0  complex, not real, answers.
F. / -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)  Use calculator: -0.6950613715. Negative exponents are acceptable.
G. / 1 / Like E above: . Write in exponential form: 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. / 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. / - 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. / / 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. / / 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. 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  x = 3 (because 81 = 34)
P. / 2 / Change to expo form  41/2 = x = = 2
Q. / / 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. / 1.292 / “Evaluate” means numerical answer. Use change-of-base formula = 1.292029674
B. / 2.262 / Change of base formula: = 2.261859507
8. / 1st ? / 11.5 yrs / 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 ? / 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
$ 2434 /  N = 24, I% = 3.5, PV = ?, Pmt = 0, FV = 3000, P/Y = C/Y = 4  PV = - 2433.974978
10. / $ 2992.73 /  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 mosquitoes / 10000 = 1000 e.588 t 10 = e.588 t ln 10 = .588 t  t = (ln 10) / .588  t = 3.917382327
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.