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Math Help - Properties of logarithms

  1. #1
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    Properties of logarithms

    Hey guys, we're starting a new lesson on properties of logs tomorrow and I'm wondering if you guys can help my understand it? I wanted to do some advanced reading and I picked 3 random examples from the textbook.

    1) ln(x+4/x-2)-2ln(x-2)

    2) 1+log312-(1/2)log318-log92

    3) 8log(b^2)√x - logb (xy)

    BTW, the symbols in smaller font are supposed to mean 'to the base of' so and so. As always, thanks for your help!
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  2. #2
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    Hello, archistrategos214!

    If you're just starting, these are awful problems!
    I assume you know the basic properties of logarithms.


    1) .ln[(x+4)/(x-2)] - 2·ln(x - 2)
    We have: .ln(x + 4) - ln(x - 2) - 2·ln(x - 2) .= .ln(x + 4) - 3·ln(x-2)

    . . = .ln(x + 4) - ln(x - 2)³ .= .ln[(x + 4)/(x - 2)³]



    2) .1 + log312 - ½·log318 - log92
    That "base 9" really messes up the problem!

    Assuming you don't know the Base-Change Formula yet,
    . . I'll show you a primitive approach.

    Let .log
    9(2) = p . . . Then: .9^p .= .2 . . (3²)^p .= .2 . . 3^{2p} .= .2

    . . Hence: .2p .= .log
    3(2) . . p .= .½·log3(2)

    Therefore: .log
    9(2) .= .½·log3(2)


    The problem becomes: .log
    3(3) + log3(12) - ½·log3(18) - ½·log3(2)

    . . = .log
    3(3·12) - ½[log3(18) + log3(2)] .= .log3(36) - ½[log3(18·2)]

    . . = .log
    3(36) - ½log3(36) .= .½·log3(36) .= .log3(36^½) .= .log3(6)

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  3. #3
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    Thanks for the help, Soroban! Yes, we're only starting this lesson today, but hopefully our instructor won't be a sadist and give us questions that are too hard. Again, thanks!
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  4. #4
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    Hello again, archistrategos214!

    I'll try to explain #3 . . . another ugly problem!


    3) .8·log(√x) - logb(xy)

    Let log
    (√x) .= .p . . (b²)^p .= .√x . . b^{2p} .= .√x

    Then: .2p .= .log
    b(x^½) . . 2p .= .½·logb(x) . . p .= .¼·logb(x)

    Hence: .log
    (√x) .= .¼·logb(x)


    The problem becomes: .8·¼·log
    b(x) - logb(xy) . = . 2·logb(x) - logb(xy)

    . . = .log
    b(x²) - logb(xy) .= .logb(x²/xy) .= .logb(x/y)

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