Question 1110675
.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The solution by @josgarithmetic is &nbsp;&nbsp;<U>W R O N G</U>.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Below find the correct solution.



<pre>
{{{log(3,(10x^2-x-2))}}} = {{{2 + 2*log(3,(x))}}}    ====>


{{{log(3,(10x^2-x-2))}}} = {{{log(3,(9)) + log(3,(x^2))}}}  <=== is equivalent ====>


{{{log(3,(10x^2-x-2))}}} = {{{log(3,(9x^2))}}}  <=== is equivalent ====>


10x^2 - x - 2 = 9x^2


x^2 - x - 2 = 0


(x-2)*(x+1) = 0


The roots to the last equation are  x= 2  and  x= -1.


The value  x= -1  does not work as the solution to the original equation,
since  {{{log(3,(-1))}}}  is not defined.


The value  x= 2  is the solution:  both the left side and the right side of the original equation are DEFINED at x= 2.


<U>Answer</U>.  The given equation has one and only one solution x= 2.


<U>Check</U>.  The left side is  {{{log(3, (10*2^2 -2 -2))}}} = {{{log(3,(36))}}} = {{{log(3,(9*4))}}} = {{{2 + 2*log(3,(2))}}}.


        The right side is  the same value.
</pre>

Solved.


----------------
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> 

&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, &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.