SOLUTION: Topic solving logarithmic solve the equation Log base 3 (10x^2-x-2)=2+2log base 3 x (Please help me do one of my assignment question)

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Question 1110675: Topic solving logarithmic
solve the equation
Log base 3 (10x^2-x-2)=2+2log base 3 x
(Please help me do one of my assignment question)
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
log%283%2C%2810x%5E2-x-2%29%29=2%2B2%2Alog%283%2Cx%29
-

The right-hand side could be rewritten
2%2Alog%283%2C3%29%2B2%2Alog%283%2Cx%29
log%283%2C3%5E2%29%2Blog%283%2Cx%5E2%29
log%283%2C9%29%2Blog%283%2C%28x%5E2%29%29
log%283%2C9x%5E2%29
.
.
.

Answer by ikleyn(52812) About Me  (Show Source):
You can put this solution on YOUR website!
.

        The solution by @josgarithmetic is   W R O N G.

        Below find the correct solution. 


log%283%2C%2810x%5E2-x-2%29%29 = 2+%2B+2%2Alog%283%2C%28x%29%29    ====>


log%283%2C%2810x%5E2-x-2%29%29 = log%283%2C%289%29%29+%2B+log%283%2C%28x%5E2%29%29  <=== is equivalent ====>


log%283%2C%2810x%5E2-x-2%29%29 = log%283%2C%289x%5E2%29%29  <=== 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%283%2C%28-1%29%29  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.


Answer.  The given equation has one and only one solution x= 2.


Check.  The left side is  log%283%2C+%2810%2A2%5E2+-2+-2%29%29 = log%283%2C%2836%29%29 = log%283%2C%289%2A4%29%29 = 2+%2B+2%2Alog%283%2C%282%29%29.


        The right side is  the same value.

Solved.

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