Question 1138795
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<pre>
The original equation is


    {{{4*log(16,(x))}}} = {{{log(4,(x+12))}}}.


Note that the domain, where both sides are defined, is the set of positive real x:  { x | x > 0 }.


Due to properties of logarithms, it is the same as

     {{{4*(1/2)*log(4,(x))}}} = {{{log(4,(x+12))}}},   or, equivalently,


     {{{2*log(4,(x))}}} = {{{log(4,(x+12))}}},   which, in turn, is equivalent to


     {{{log(4,(x^2))}}} = {{{log(4,(x+12))}}}.


It implies

      x^2 = x + 12

      x^2 - x - 12 = 0,

      (x-4)*(x+3) = 0.


Of the two roots, x= 4 and x= -3, only positive x= 4 is in the domain and is, therefore, the solution.


<U>ANSWER</U>.  x= 4.
</pre>

Solved.  &nbsp;&nbsp;//  &nbsp;&nbsp;I mean, &nbsp;solved in a way as it should be done.


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On the way, &nbsp;I used standard and elementary properties of logarithms that every student must learn and must know 
before starting solving such problems.


On logarithms and their properties, &nbsp;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> 

in this site.


Also, &nbsp;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>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.