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Math Help - Exponential function properties

  1. #1
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    Exponential function properties

    I am learning about exponential function so I need some help to understand.

    I am learning about properties of function f(x) = 2^x ,x \in Q

    I need explanation of one property.

    Property is: for all x_1 ,x_2  \in Q,x_1  < x_2  \Rightarrow 2^{x_1 }  < 2^{x_2 }

    Proof of this property is:
    let x_1  = \frac{{p_1 }}{{q_1 }},x_2  = \frac{{p_2 }}{{q_2 }},p_1 ,p_2  \in Z,q_1 ,q_2  \in N
    let n be common factor of numbers q_1,q_2 so we have x_1  = \frac{{m_1 }}{n},x_2  = \frac{{m_2 }}{n}...

    I need explanation how did x_1  = \frac{{p_1 }}{{q_1 }},x_2  = \frac{{p_2 }}{{q_2 }} become x_1  = \frac{{m_1 }}{n},x_2  = \frac{{m_2 }}{n}?
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by DenMac21
    I am learning about exponential function so I need some help to understand.

    I am learning about properties of function f(x) = 2^x ,x \in Q

    I need explanation of one property.

    Property is: for all x_1 ,x_2  \in Q,x_1  < x_2  \Rightarrow 2^{x_1 }  < 2^{x_2 }

    Proof of this property is:
    let x_1  = \frac{{p_1 }}{{q_1 }},x_2  = \frac{{p_2 }}{{q_2 }},p_1 ,p_2  \in Z,q_1 ,q_2  \in N
    let n be common factor of numbers q_1,q_2 so we have x_1 = \frac{{m_1 }}{n},x_2 = \frac{{m_2 }}{n}...

    I need explanation how did x_1 = \frac{{p_1 }}{{q_1 }},x_2 = \frac{{p_2 }}{{q_2 }} become x_1 = \frac{{m_1 }}{n},x_2 = \frac{{m_2 }}{n}?
    I'm not sure why someone would want to use this method to do the proof, but in general, your m_1 and m_2 are no longer integers. If q_1=an then m_1=p_1/a for example.

    -Dan
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  3. #3
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    Quote Originally Posted by topsquark
    I'm not sure why someone would want to use this method to do the proof, but in general, your m_1 and m_2 are no longer integers. If q_1=an then m_1=p_1/a for example.

    -Dan
    Let me write complete proof:

    Property is: for all x_1 ,x_2  \in Q,x_1  < x_2  \Rightarrow 2^{x_1 }  < 2^{x_2 }

    Proof of this property is:
    let x_1  = \frac{{p_1 }}{{q_1 }},x_2  = \frac{{p_2 }}{{q_2 }},p_1 ,p_2  \in Z,q_1 ,q_2  \in N
    let n be common factor of numbers q_1,q_2 so we have q_1  = an,q_2  = bn so x_1  = \frac{{p_1 }}{{an}} = \frac{{p_1 }}{a}*\frac{1}{n} = \frac{{m_1 }}{n},x_2  = \frac{{p_2 }}{{bn}} = \frac{{p_2 }}{b}*\frac{1}{n} = \frac{{m_2 }}{n}
    x_1  = \frac{{m_1 }}{n},x_2  = \frac{{m_2 }}{n}

    if x_1<x_2 then m_1<m_2 and because of that
    2^{x_1 }  = 2^{\frac{{p_1 }}{{q_1 }}}  = 2^{\frac{{m_1 }}{n}}  < 2^{x_2 }  = 2^{\frac{{p_2 }}{{q_2 }}}  = 2^{\frac{{m_2 }}{n}} which we wanted to prove.
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