Question 172610
Hi, Hope I can help,
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solve for x 
{{{ log(4,(x-6))+log(4,x)=2 }}}
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If you add to of the same logs together, that is the same thing as multiplying the terms together, with only one log, this is what I mean
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{{{ (x-6)(x) }}} = {{{ x^2 - 6x }}}
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The new log would be
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{{{ log(4,(x^2 - 6x))=2 }}}
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From here we will change this logarthimic equation into an exponential equation
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This is how you change a logarthmic equation ( logs are a power of a base )
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{{{ log ( 3, 243 ) = 5 }}} = {{{ 3^5 = 243 }}}
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For our equation
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{{{ log(4,(x^2 - 6x))=2 }}} = {{{ 4^2 = x^2 - 6x }}} = {{{ 16 = x^2 - 6x }}}
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Now we can move the "16" to the right and solve the quadratic equation
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{{{ 16 = x^2 - 6x }}} = {{{ 16 - 16 = x^2 - 6x - 16 }}} = {{{ 0 = x^2 - 6x - 16 }}} or {{{ x^2 - 6x - 16 = 0}}}
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{{{ x^2 - 6x - 16 = 0}}}, we can factor this equation,
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( x )(x ), first name all the factors of (-16)
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One of these pairs will have to add up to (-6)
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({{{ -16 }}}, {{{ 1 }}}) ( added will equal (-15) )
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({{{ 16 }}}, {{{ -1 }}}) ( added will equal "15" )
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({{{ (8) }}}, {{{ -2 }}}) ( added will equal "6") 
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({{{ highlight(-8) }}}, {{{ highlight(2) }}}) ( added will equal (-6))
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You put the highlighted factors into the parentheses with the "x"'s
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{{{ ( x - 8)(x + 2 ) = 0 }}}, you can check by using the foil method, I checked and it is the same as {{{ x^2 - 6x - 16 }}}
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You can find "x" by placing both factors equal to "0"
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{{{ ( x - 8) = 0 }}} = {{{ x - 8 = 0 }}}, move the (-8) over to the right side
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{{{ x - 8 = 0 }}} = {{{ x - 8 + 8 = 0 + 8 }}}, {{{ x = 8 }}}
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or
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{{{ (x + 2 ) = 0 }}} = {{{ x + 2  = 0 }}}, move the "2" to the right side
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{{{ x + 2  = 0 }}} = {{{ x + 2 - 2 = 0 - 2 }}}, {{{ x = (-2) }}}
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{{{ x = 8 }}} and  {{{ x = (-2) }}}, the only thing left is to put these values into the original equation, logs can't be nagative, so we need to make sure that our answers don't create negative numbers
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(x = 8), {{{ log(4,(x-6))+log(4,x)=2 }}} = {{{ log(4,(8-6))+log(4,8)=2 }}} = {{{ log(4,2)+log(4,8)=2 }}} ( doesn't make negative, so "8" is a good answer
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(x = (-2)), {{{ log(4,(x-6))+log(4,x)=2 }}} = {{{ log(4,(-2-6))+log(4,-2)=2 }}} = {{{ log(4,(-8))+log(4,-2)=2 }}} ( it creates a negative number, so (-2) is no good)
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The only answer that works is {{{ 8 }}}, {{{ x = 8 }}}
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The graph of this equation is
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{{{ graph ( 400,400,-10,10,-10,10, log(4,(x^2 - 6x))-2) }}}
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Remember though that "x" can only equal "8", "x" can't be equal to (-2) since this answer made the logs negative
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{{{ x = 8 }}}
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Hope I helped, Levi