Expert: Sherman D. - 3/19/2005

Question
17. Evaluate the expression

 3^(-2 pi)

18. A virus accidently brought to a remote village with a population of 7500 that has never been exposed to te disease. The spread of the virus is modeled by
  y= 7500/ 1+7499e^(-0.6t)

where t is the time in days since the virus was introduced. How many villagers will be infected after 5 days.

20. Write the logarithmic expression in exponential form.

 log(256) 64= 3/4

21. Evaluate the expression
 3500(3.19)

22. Find the value of x
 3 ln (4x) = 13

24. Evaluate the logarithm using the change of base formula.

 log(5) 786

25. The magnitude of an earthquake as measured by the richter scale is

R= 0.67 log(10) [0.37e] + 1.46

Where R is the magniture of the earthquake and e is the energy in kilowatt hours released by the earthquake. Which magnitude corresponds to a release of 1.46 * 10^6 KWH of energy

27. Use the properties of logarithms to expand the expression (assume variables are positive)

  log(4) x^4 * sqrt 4

28. The number of bacteria present in a culture is

B= 1000e^(0.276t)

Where t is the time in minutes. Find the time required to the nearest half minute, to have 6000 bacteria present

30. Find the value of the expression w/o using a calculator

3 log(3) sqrt 3

Answer
17.) Evaluate the expression

3^(-2 pi)
this is the same as saying
3^2^-pi
becomes
(3^2)^(-pi)
9^(-pi)

you can either write this as
9^(-pi) or (1/9)^pi either one is exceptable.

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18.) A virus accidently brought to a remote village with a population of 7500 that has never been exposed to te disease. The spread of the virus is modeled by
y = 7500/ 1+7499e^(-0.6t)

where t is the time in days since the virus was introduced. How many villagers will be infected after 5 days.

In short, if you substitute 5 in for t, you get 20 villagers as the answer. For a better explanation read on.

y = 7500/(1 + 7499e^(-.6 * 5)
y = 7500/(1 + 7499e^(-3))
y = 7500/(1 + (7499/(e^3)))
y = 7500/((e^3 + 7499)/e^3)
y = (7500/1)/((e^3 + 7499)/e^3))
y = (7500/1)*((e^3)/(e^3 + 7499))
y = (7500e^3)/(e^3 + 7499)
y = 20.03455423
since you can't have a fraction of a villager
y = 20 villagers that will be infected after 5 days.

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20.) Write the logarithmic expression in exponential form.

log(256)64 = 3/4
256^(3/4) = 64

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21.) Evaluate the expression
3500(3.19)

ANS : 11165

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22.) Find the value of x
3ln(4x) = 13

ln((4x)^3) = 13
ln(64x^3) = 13
inverse ln both sides
64x^3 = 442413.392
divide both sides by 64
x^3 = 6912.70925
cbrt both sides
x = 19.04946414
x = about 19.05

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Just in case you don't know, you left out #23 just in case you didn't mean to.

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24.) Evaluate the logarithm using the change of base formula.

log(5)786 = (log(786))/(log(5))

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25.) The magnitude of an earthquake as measured by the richter scale is

R = 0.67 log(10) [0.37e] + 1.46

Where R is the magniture of the earthquake and e is the energy in kilowatt hours released by the earthquake. Which magnitude corresponds to a release of 1.46 * 10^6 KWH of energy

If by this you mean

R = .67(log(10))(.37e) + 1.46
then
R = (log((.37e)^(.67)))/(log(10)) + 1.46
R = .001672457996 + 1.46
R = 1.461672458

If you meant to type it another way, then just let me know.

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27.) Use the properties of logarithms to expand the expression (assume variables are positive)

If by this you mean

log(4)(x^4 * sqrt 4)
log(4)(x^4) + log(4)(sqrt(4))
4log(4)x + log(4)(4^(1/2))
4log(4)x + (1/2)log(4)4

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28.) The number of bacteria present in a culture is

B = 1000e^(0.276t)

Where t is the time in minutes. Find the time required to the nearest half minute, to have 6000 bacteria present


6000 = 1000e^(.276t)
divide both sides by 1000
6 = e^(.276t)
log(e)6 = .276t
ln(6) = .276t
divide both sides by .276
(ln(6))/.276 = t
t = 6.491882135
t = 6 minutes .491882135
or
t = about 6 minutes and 30 seconds
in short
t = about 6.5 minutes

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30.) Find the value of the expression w/o using a calculator

3log(3)(sqrt(3))

log(3)((sqrt(3))^3)
log(3)(3^(3/2))
using that, in exponential form, you get
3^x = 3^(3/2)
as you can see the answer is 1.5 or (3/2)