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Answers to the sample test!!

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1)  Simplify each:
                     a)  4 ^. 2^-4 =                     b)  (4 ^. 2)^-4 =              c)  (a^-1 + b^-1)^-2 =
                                       4(1/16)                           = 1/8⁴                         (1/a + 1/b)^-2
                            = 1/4                            =  1/4096                   = [(a + b)/ab]^-2
                                                                                                   =[ab/(a + b)]²
                                                                                                  = a²b²/(a + b)²

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2)  Simplify each:
 
 
  a)  (25/9)^1/2 = 5/3                  b)  16^1/2 = 4                    c)  x - 2

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3)  Solve the equation:  9^3x = 81^{x + 1}
                                                          9^3x = 9^{2(x + 1)
                                                3x = 2(x + 1)
                                        3x = 2x + 2
                                           x = 2}

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^{4)  The half life of a radioactive isotope is 7 days.  If 5.6 kg are present now, how much will be present in t days?  In 13 days?}
  A(t) = 5.6(1/2)^t/7
A(13) = 5.6(1/2)^13/7 = 1.55 kg

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^{5)  A bacteria colony triples every 6 days.  The population currently is 5000 bacteria.  What will be the population in 21 days?}
  A(21) = 5000(3)^21/6 = 233827

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^{6)  $1000 is invested at 8% compounded quarterly for 3 years.  How much interest will you receive at the end of the time period?}
  A(3) = 1000(1 + .08/4)¹² = $1268.24

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^{7)  You invest $3000 at 7% compounded continuously for 2 years.  How much money will be in the account at the end of the time period?}
  P(2) = 3000 e^.14 = $3450.82

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8)  Solve each log equation:
                 a)  Log x = 21
            Write exponentially,           10²¹ = x

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                 b)  Log |x| = 15
                        Write exponentially and solve: | x | = 10¹⁵
                                                                                                            x = 10¹⁵ or -10¹⁵

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                 c)  Ln (x² - 1) = 3
                                Write exponentially and solve e³ = x² - 1 e³ + 1 = x²                                                           _____ + \/ e³ + 1  = x

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                   d)  Log x = 1.6  (Use calculator and round to hundredths.)
  Write exponentially,  10^1.6 = x
                                     x = 39.81

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  9)  Write each log as a single log:
 
                 a)  Log x + log y + 2Log z
                                              =Log xyz²

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                 b)  Ln x + Ln y - 3Ln z
                          =  Ln (xy/z³)

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10)  Simplify each log:
 
                 a)  Ln e⁵ =     5
 
                 b)  10^2log4 =   16
 
                 c)  10^{1 + log 5} = (10)(10^{log 5}) = (10)5 = 50

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  11)  Solve the equation:  Log (x + 2) + Log 5 = 4 Log 5(x + 2) = 4 5(x + 2) = 10⁴ 5x + 10 = 10000 5x = 9990 x = 1998

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12)  Graph y = 3^x and y = log₃ x on the same axis.   

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13)  Use a calculator to find Log₇ 58 and round to hundredths place.
  = log 58/log 7 = 2.09

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14)  Solve the equation:  15^{x + 1} = 29 and round to hundredths place.   (x + 1)log 15 = log 29 x log 15 + log 15 = log 29 xlog 15 = log 29 - log 15 x = (log 29 - log 15)/ log 15 x = .24

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15)  How long will it take to double $1000 if invested at 6% compounded monthly?   2000 = 1000(1 + .06/12)^12t 2 = (1 + .06/12)^12t log 2 = 12t log (1 + .06/12) log 2/ 12log(1 + .06/12)  = t 11.58 = t 11.58 years

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