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Math Help - Logarithms

  1. #1
    Member rowe's Avatar
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    Logarithms

    We have a function such that f(t) = 10 \cdot 2^{kt} and f(\tfrac{1}{2}) = 3. Find k.

    f(\tfrac{1}{2}) = 3, therefore:

    3 = 10 \cdot 2^{\tfrac{k}{2}}

    Taking the log to the base 2 on both sides:

    log_2 3 = log_2 10 + \frac{k}{2}

    Now, I know that log_a xy = log_a x + log_a y, but this must mean that we had:

    log_2 3 = log_2 10 + log_2 2^{\tfrac{k}{2}}

    Can someone show me why (proof) that log_2 2^{\tfrac{k}{2}} = \frac{k}{2}?


    And is it just me, or do those log renders look a bit squashed? The logs and the fractional exponent. Is there a better way to render these?
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  2. #2
    RRH
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    <br /> <br />
3 = 10 \cdot 2^{\tfrac{k}{2}}<br /> <br />

    Divide both sides by ten ... it should now look like this

    <br /> <br />
    \frac{3}{10}= 2^{kt}<br /> <br />

    Now

    <br /> <br />
    log 3 - log 10 = \frac{k} {2} log 2<br /> <br />

    multiply both sides by 2

    <br /> <br />
2(log 3 - log 10) = k log 2<br /> <br />

    Divide both sides by log 2 you have now isolated k





    Sorry about this being sloppy I am not very good at using Latex
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  3. #3
    Member rowe's Avatar
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    Thanks for the solution, but I am interested in finding out why first:

    log_2 2^{\tfrac{k}{2}} = \frac{k}{2}

    I'm not familiar with the proof or general rule for why this is.
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  4. #4
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by rowe View Post
    Thanks for the solution, but I am interested in finding out why first:

    log_2 2^{\tfrac{k}{2}} = \frac{k}{2}

    I'm not familiar with the proof or general rule for why this is.
    If you're happy with log_c(ab) = log_c(a)+log_c(b) then that can be used (although I'm sure someone can make it more elegant

    x^n = x \cdot x \cdot x ... up to n times

    log_2(x \cdot x \cdot x ... \cdot x)

    = log_2(x) + log_2(x) + log_2(x) + ... + log_2(x)

    Factor out like terms (and there are n of them)

    n(log_2(x))

    Wikipedia defines it in terms of the inverse (exponents): http://en.wikipedia.org/wiki/List_of...ler_operations
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  5. #5
    RRH
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    Here is a website with the rules for logarithms it may be helpfull

    RULES OF LOGARITHMS
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