Expert: Sherman D. - 6/10/2009

Question
22. Express as a single logarithm:
ln 7 + 0.5 ln 9  

23. Express as a single logarithm:
4 ln 2 - ln 3 - 1

24. Solve:
ln1/x = 2


25. Solve:
ln|x| = 1  

26. Solve:
ln sq root x = 3


27. Solve:
e^-x = 3

28. Solve:
e^3x = 8


29. Solve:
1/ex = 7


30. Solve:
e^-2x = 0.2

31. Solve:
3e^2x + 2 = 50  

32. Solve:
|ln x| = 1

33. Solve:
2 ln x = ln(x + 1)

34. Capital Trust offers a savings accounting with 6% interest compounded monthly, while South Market offers a savings account with 6% interest compounded quarterly. Which account is better and why?

35. Solve (You may leave your answer in terms of ln or e):
ln(lnx)=0

36. Solve (You may leave your answer in terms of ln or e):
lnx+ln(x+3) = ln10  

Answer
22. Express as a single logarithm:
ln 7 + 0.5 ln 9  

ln(7) + (1/2)ln(9)
ln(7) + ln(9^(1/2))
ln(7) + ln(3)
ln(7*3)
ln(21)

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23. Express as a single logarithm:
4 ln 2 - ln 3 - 1

4*ln(2) - ln(3) - 1
ln(2^4) - ln(3) - 1
ln(16) - ln(3) - ln(e)

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24. Solve:
ln1/x = 2

ln(1/x) = 2
ln(x^-1) = 2
-1*ln(x) = 2
ln(x) = -2
x = e^(-2) or 1/(e^2)

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25. Solve:
ln|x| = 1  

since you have an absolute value mark around the x

ANS : -e or e

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26. Solve:
ln sq root x = 3

ln(sqrt(x)) = 3
(1/2)ln(x) = 3
ln(x) = 6
x = e^6

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27. Solve:
e^-x = 3

e^(-x) = 3
-x = ln(3)
x = -ln(3) or ln(1/3)

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28. Solve:
e^3x = 8

e^(3x) = 8
3x = ln(8)
x = (1/3)ln(8)
x = ln(8^(1/3))
x = ln(2)

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29. Solve:
1/ex = 7
1/(ex) = 7
7ex = 1
ex = 1/7
x = 1/(7e)
but if you meant
(1/e)x = 7
x = 7e

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30. Solve:
e^-2x = 0.2

e^(-2x) = .2
-2x = ln(.2)
x = (-1/2)ln(.2)
x = ln((1/5)^(-1/2))
x = ln(5^(1/2))
x = (1/2)ln(5)

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31. Solve:
3e^2x + 2 = 50  

if by this you mean

3e^(2x) + 2 = 50
3e^(2x) = 48
e^(2x) = 16
2x = ln(16)
x = (1/2)ln(16)
x = ln(16^(1/2))
x = ln(4)

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32. Solve:
|ln x| = 1

this can be rewritten as

ln(x) = +/- 1
x = 1/e or e

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33. Solve:
2 ln x = ln(x + 1)

2*ln(x) = ln(x + 1)
ln(x^2) = ln(x + 1)
x^2 = x + 1
x^2 - x - 1 = 0

x = (-b +/- sqrt(b^2 - 4ac))/(2a)

x = (1 +/- sqrt(1 + 4))/2
x = (1 +/- sqrt(5))/2

since you can't use a negative number

ANS : x = (1/2)(1 + sqrt(5))

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34. Capital Trust offers a savings accounting with 6% interest compounded monthly, while South Market offers a savings account with 6% interest compounded quarterly. Which account is better and why?

P = C(1 + (r/n))^(nt)
The Only parts that matter oare (1 + (r/n))^n

P1 = (1 + (.06/12))^(12) = 1.0617
P2 = (1 + (.06/4))^(4) = 1.0202

So the first account is better.

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35. Solve (You may leave your answer in terms of ln or e):
ln(lnx)=0

ln(ln(x)) = 0
ln(x) = e^0
ln(x) = 1
x = e^1
ANS : x = e

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36. Solve (You may leave your answer in terms of ln or e):
ln(x) + ln(x + 3) = ln(10)
ln(x(x + 3)) = ln(10)
x^2 + 3x = 10
x^2 + 3x - 10 = 0
(x + 5)(x - 2) = 0
x = -5 or 2
since you can't do ln(-5) because that would make your problem invalid
ANS : x = 2