SOLUTION: sOLVE log x(2x-1)=1

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Question 1092571: sOLVE
log x(2x-1)=1
Answer by ikleyn(52910) About Me  (Show Source):
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highlight%28PLEASE%29 solve
log x(2x-1)=1
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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%5B1%2C2%5D = %281+%2B-+sqrt%281%5E2+%2B+4%2A2%2A10%29%29%2F%282%2A2%29  = %281+%2B-+sqrt%2881%29%29%2F4 = %281+%2B-+9%29%2F4.


x%5B1%5D = %281%2B9%29%2F4 = 10%2F4 = 2.5.  It works:  log (2.5*(2*2.5-1)) = log (2.5*4) = log (10) = 1.

x%5B2%5D = %281-9%29%2F4 = -8%2F4 = -2.    It works too, since log ((-2)*(2*(-2)-1) = log ((-2)*(-5)) = log(10) = 1.


Answer.  The given equation has two roots x= 2.5  and  x= -2.




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


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On logarithms, their properties, and logarithmic equations see the lessons
    - WHAT IS the logarithm
    - Properties of the logarithm
    - Change of Base Formula for logarithms
    - Solving logarithmic equations
    - Using logarithms to solve real world problems
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Logarithms".


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.