Exponential function properties

**DenMac21** · February 18th 2006, 04:18 AM

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 $x_1 = \frac{{m_1 }}{n},x_2 = \frac{{m_2 }}{n}$ ?

**topsquark** · February 18th 2006, 05:05 AM

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

**DenMac21** · February 18th 2006, 10:36 AM

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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