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Math Help - The real exponential function

  1. #1
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    The real exponential function

    I'm stuck on this question, for two reasons. Firstly, I don't know what it's asking, and secondly, because I doubt I can show what is being asked.

    1) Show that there is at least one function exp: → R^+ so that
    i) exp(x+y) = exp(x)*exp(y), x and y ∈ R
    ii) 1 + X <= exp(x), x ∈ R
    2) Using i) and ii) show that the function exp is monotonic strictly increasing.

    Help would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by Olym View Post
    I'm stuck on this question, for two reasons. Firstly, I don't know what it's asking, and secondly, because I doubt I can show what is being asked.

    1) Show that there is at least one function exp: → R^+ so that
    i) exp(x+y) = exp(x)*exp(y), x and y ∈ R
    ii) 1 + X <= exp(x), x ∈ R
    2) Using i) and ii) show that the function exp is monotonic strictly increasing.

    Help would be greatly appreciated.
    What facts are you allowed to use? Certainly, defining exp(x)= 2^x has the properties in (1) doesn't it? And, if so, then you have shown that that function, at least, has those properties.

    For 2, assume y> 0 and look at exp(x+ y)= exp(x)*exp(y)> exp(x)(1+y). Do you see how that shows that exp(x+y)> exp(x)? Do you see why that says that exp(x) is an increasing function?
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  3. #3
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    Quote Originally Posted by HallsofIvy
    What facts are you allowed to use? Certainly, defining exp(x)= 2^x has the properties in (1) doesn't it? And, if so, then you have shown that that function, at least, has those properties.
    The question makes no mention of what properties are allowed. The question only hints that it's enough to examine the function in the interval of [0, 1]. Another hint given was " exp(x)*exp(-x) = ?".

    Quote Originally Posted by HallsofIvy
    For 2, assume y> 0 and look at exp(x+ y)= exp(x)*exp(y)> exp(x)(1+y). Do you see how that shows that exp(x+y)> exp(x)? Do you see why that says that exp(x) is an increasing function?
    No, sorry, I don't. Could you explain some more?

    Thanks
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    Certainly, defining exp(x)= 2^x has the properties in (1) doesn't it?
    Actually, no it does not.
    \forall x\in (0,1),~2^x<x+1
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  5. #5
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    Quote Originally Posted by Olym View Post
    I'm stuck on this question, for two reasons. Firstly, I don't know what it's asking, and secondly, because I doubt I can show what is being asked.

    1) Show that there is at least one function exp: → R^+ so that
    i) exp(x+y) = exp(x)*exp(y), x and y ∈ R
    ii) 1 + X <= exp(x), x ∈ R
    2) Using i) and ii) show that the function exp is monotonic strictly increasing.

    Help would be greatly appreciated.
    Hi,

    I can't believe you are asked this question without a few prior questions. Perhaps your teacher has constructed a family of functions, and you have to use this construction, or your teacher told you to read some book about this, I don't know...

    The result of question 1) took centuries to mathematicians before it was established (long ago), you can't find it out of nowhere... Of course now it has become an easy question, for those who know the answer. If you know some about series (mainly, you need the Cauchy product of two series), then the easiest way is to define \exp(x)=\sum_{n=0}^\infty \frac{x^n}{n!} (provided you prove this makes sense) and check i) and ii). But once again, you should first read your last lecture notes, it probably gives the way your teacher wants you to answer.
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