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Core 3 help

  1. Jun 24, 2007 #1
    Hi.

    Could someone please help me with the following questions? I'm totally stuck so any feedback would be really appreciated. I'll post the questions one at a time.

    Express as single log. functions:

    (i) ln (x + 1) - 3 ln (1 - x) + 2 ln x


    I got up to ln ( (x + 1) / (1 - x)^3 ) / ln x^2 but I don't know what to do next or whether this is even correct.

    Thank you.

    Cathy
     
  2. jcsd
  3. Jun 24, 2007 #2

    Astronuc

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    Staff: Mentor

    Using the identity ln(xy) = ln x + ln y, one can extend it to ln (xyz) = ln x + ln y + ln z, and ln (x/y) = ln x - ln y.

    So in the problem ln (x + 1) - 3 ln (1 - x) + 2 ln x

    one obtains ln [(x+1)/(1-x)3] + ln x2 which can be further simplified by bringing x2 inside the logarithm operation.

    ln [(x+1)x2/(1-x)3]

    Reference: http://mathworld.wolfram.com/Logarithm.html
     
  4. Jun 24, 2007 #3
    Thanks so much for your help.

    Cathy
     
  5. Jun 24, 2007 #4
    Could someone please give me a tip for how to answer this one as I've no idea what to do? I'd really appreciate it.

    Given ln (xy^3) = m and ln (x(^3)y(^2)) = n, find ln root(xy) in terms of m and n.

    Cathy
     
  6. Jun 24, 2007 #5

    Astronuc

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    Staff: Mentor

    By root (xy), does one mean [itex] \sqrt{xy} [/itex]?

    If so, then ln (xy)1/2 = 1/2 ln xy = 1/2 (ln x + ln y)

    and one also needs to exand the equations ln (xy3) = ln x + 3 ln y = m, and similarly for the other equation, then rearrage to x and y in terms of m and n.
     
    Last edited: Jun 24, 2007
  7. Jun 24, 2007 #6
    Yeah, that's what I meant. I just don't understand how to connect that to m and n.
     
  8. Jun 24, 2007 #7
    Oh, I've got that one now.

    I have one last query and that is how to draw the graphs (and asymptotes) of y = 2 + ln x and y = - ln (x - 3).

    Thank you.

    Cathy
     
  9. Jun 24, 2007 #8

    Astronuc

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    Staff: Mentor

    Well, certain as x gets very large, y = (2 + ln x) ~ ln x,

    and similarly as x gets very large, i.e. x >> a, then x+a ~ x.

    Also, think of the range for ln (3-x).
     
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