Question 1118244
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Compare 10^2 and 2^10. For any two numbers make a conjucture about which usually gives the greater number, as the base or as the exponent.
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I'd like to answer the second question of this post.


<pre>
    For any two numbers make a conjecture about which usually gives the greater number, as the base or as the exponent.
</pre>


So, &nbsp;the question is:  &nbsp;&nbsp;if &nbsp;x &nbsp;and &nbsp;y &nbsp;are two positive numbers  and  &nbsp;&nbsp;y > x,  &nbsp;what is greater,  &nbsp;&nbsp;{{{x^y}}}  &nbsp;or  &nbsp;{{{y^x}}} ?


The answer is:  &nbsp;if both  &nbsp;x  &nbsp;and  &nbsp;y &nbsp;are greater than &nbsp;"e"  &nbsp;and  &nbsp;&nbsp;x < y,  &nbsp;then  &nbsp;&nbsp;{{{x^y}}} > {{{y^x}}}.     &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*)


<pre>
    In wording form: if in the domain  x, y > e  you raise the smaller number to the greater degree, you will get the greater number 
    than if you raise the greater number to the smaller degree.
</pre>


<U>Solution</U>


<pre>
Let's start from this inequality

    {{{x^y}}} > {{{y^x}}}.               (1)


Take the logarithm from both sides. You will get


    y*ln(x) > x*ln(y).      (2)

Divide both sides by the product  ln(x)*ln(y).  You will get

    {{{y/ln(y)}}} > {{{x/ln(x)}}}.            (3)           <<<---=== in the domain  x, y > e you are safe to do it . . . 


Now the <U>KEY STATEMENT</U> is that the inequality (3) is TRUE (!).    


Why it is true ?  - Because the function  F(x) = {{{x/ln(x)}}}  monotonically increases as  x  increases in the domain  x > e.


    I will not prove the last statement.


    Everybody who is familiar with Calculus, can easily do it on his or her own.


    Which is better, I will demonstrate this fact visually, by presenting the plot:



    {{{graph( 330, 330, -1.5, 10.5, -5.5, 10.5,
          x/ln(x)
)}}}


    Plot y = {{{ln(x)/x}}}


Now, if you reverse the arguments in this sequence  (3)  ---->  (2)  ---->  (1), you will get the major statement  (*)  <U>proved</U>.
</pre>

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So, &nbsp;to the popular question - which number is greater,  &nbsp;&nbsp;{{{e^pi}}}  &nbsp;or  &nbsp;&nbsp;{{{pi^e}}},  &nbsp;&nbsp;the answer is:  &nbsp;&nbsp;{{{e^pi}}} &nbsp;&nbsp;is greater than  &nbsp;&nbsp;{{{pi^e}}}.