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Math Help - log

  1. #1
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    log

    1. if a car depreciates at 15% per year, how long until it is worth half its original value?

    2. led zepellin once played a concert with exerted sounds of 10 000 times the intensity of a sound that can just be heard. what is the loudness in decibels?
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  2. #2
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    Quote Originally Posted by william View Post
    1. if a car depreciates at 15% per year, how long until it is worth half its original value?
    [snip]
    Hi William,

    If a Car has value 100 today, in 1 year's time it will depreciate by 15% i.e. 100 * 0.15, So:

    today 100
    year 1 100 - 100 * 0.15
    year 2 (100 - 100 * 0.15) - (100 - 100 * 0.15)* 0.15
    etc

    you can see the pattern: Where A is the present value and r is the rate
    A(1 - r) - A(1 - r).r = A(1- r) [1 - r] = A(1 - r)(1 - r)

    So in general the depreciation is:

    A(1 - r)^{y-1} where y is the period in years

    So:

    A(1 - r) ^{y-1} = A / 2 \: :\(to reduce the value by one half).

    (1 - r)^{y-1} = 0.5

    (y - 1) log (1 - r) = log(0.5)

    y = 1 + \frac{log(0.5)}{log(0.85)}

    period = 5.26 years
    Last edited by mr fantastic; July 25th 2009 at 06:00 PM. Reason: Added quote
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  3. #3
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    Quote Originally Posted by s_ingram View Post
    Hi William,

    If a Car has value 100 today, in 1 year's time it will depreciate by 15% i.e. 100 * 0.15, So:

    today 100
    year 1 100 - 100 * 0.15
    year 2 (100 - 100 * 0.15) - (100 - 100 * 0.15)* 0.15
    etc

    you can see the pattern: Where A is the present value and r is the rate
    A(1 - r) - A(1 - r).r = A(1- r) [1 - r] = A(1 - r)(1 - r)

    So in general the depreciation is:

    A(1 - r)^{y-1} where y is the period in years

    So:

    A(1 - r) ^{y-1} = A / 2 \: :\(to reduce the value by one half).

    (1 - r)^{y-1} = 0.5

    (y - 1) log (1 - r) = log(0.5)

    y = 1 + \frac{log(0.5)}{log(0.85)}

    period = 5.26 years

    Why do you have y-1 as the exponent is your equation. By doing that, if you let y=1 which would be 1 year, your formula would just yield A, meaning no depreciation took place
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  4. #4
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    Hi Artvandalay11,

    your right, it should just by y not y-1. That's the problem when you try answering questions at 2.00 am! Thanks for the correction.
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