Question 1080916
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Please help me solve this problem:

If xy = 64 and {{{ log (x, y) + log (y, x) = 5/2 }}} find x and y.
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<pre>
{{{log (x, y) + log (y, x)}}} = {{{5/2 }}}.      (1)

Well known rule for the base change of logarithms says that  {{{log (y, x)}}} = {{{1/log(x,y)}}}.

Therefore, the equation (1) becomes

{{{log (x, y)}}} + {{{1/log (x, y)}}} = {{{5/2 }}}.   (2)

To solve it, introduce new variable  z = {{{log (x, y)}}}.  Then the equation (2) becomes

{{{z + 1/z}}} = {{{5/2}}},   or   {{{z^2 - (5/2)z + 1}}} = 0,   or   {{{2z^2 - 5z + 2}}} = 0.

Use the quadratic formula to find its roots. The roots are  {{{z[1]}}} = 2   and/or  {{{z[2]}}} = {{{1/2}}}.


1.  {{{z[1]}}} = 2  ====>  {{{log(x,y)}}} = 2  ====>  y = x^2. 

    Substitute it into the equation  xy = 64,  and you will get  {{{x^3}}} = 64,   which implies  x = 4.

    In this case, the solution of the original system is  x = 4,  y = 16.


2.  {{{z[2]}}} = {{{1/2}}}  ====>  {{{log(x,y)}}} = {{{1/2}}}  ====>  y = x^(1/2). 

    Substitute it into the equation  xy = 64,  and you will get  x^(3/2) = 64,   which implies  {{{x^3}}} = {{{64^2}}}  and hence  x = 16.

    In this case, the solution of the original system is  x = 16,  y = 4.
</pre>

<U>Answer</U>. &nbsp;There are two solutions:  &nbsp;1) x=4, y= 16,  &nbsp;and  &nbsp;2) x=16, y = 4.


Solved.



For properties of logarithms and solving logarithmic equations see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/logarithm/what-is-the-logarithm.lesson>WHAT IS the logarithm</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/logarithm/Properties-of-the-logarithm.lesson>Properties of the logarithm</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/logarithm/change-of-base-formula-for-logarithms.lesson>Change of Base Formula for logarithms</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/logarithm/How-to-solve-logarithmic-equations.lesson>Solving logarithmic equations</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/logarithm/Using-logarithms-to-solve-real-world-problems.lesson>Using logarithms to solve real world problems</A> 

in this site.


Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Logarithms</U>".