Question 1092571
.
{{{highlight(PLEASE)}}} solve

log x(2x-1)=1
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<pre>
log x(2x-1) = 1      (by default, the base of the logarithm assumes to be 10)  ====>

x*(2x-1) = 10  ====>

2x^2 - x - 10 = 0  ====>

{{{x[1,2]}}} = {{{(1 +- sqrt(1^2 + 4*2*10))/(2*2)}}}  = {{{(1 +- sqrt(81))/4}}} = {{{(1 +- 9)/4}}}.


{{{x[1]}}} = {{{(1+9)/4}}} = {{{10/4}}} = 2.5.  It works:  log (2.5*(2*2.5-1)) = log (2.5*4) = log (10) = 1.

{{{x[2]}}} = {{{(1-9)/4}}} = {{{-8/4}}} = -2.    It works too, since log ((-2)*(2*(-2)-1) = log ((-2)*(-5)) = log(10) = 1.


<U>Answer</U>.  The given equation has two roots x= 2.5  and  x= -2.
</pre>


{{{graph( 330, 330, -5.5, 5.5, -3.5, 3.5,
          log(10, (x*(2x-1))), 1
)}}}


Plot y = log (x*(2x-1)) (red), y = 1 (green)



-------------------------
On logarithms, their properties, and 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, &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.