Question 743558
{{{ 2*log(4,x) - log(4, 3x - 4) = 1/2 }}}
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The key is to make the substitution
{{{ 1/2 = log( 4,2 ) }}}
{{{ log(4,x^2) - log(4, 3x - 4) = log( 4,2 ) }}}
{{{ log( 4, (x^2 / ( 3x - 4)) ) = log( 4,2 ) }}}
{{{ x^2 / ( 3x-4 )  = 2 }}}
{{{ x^2 = 2*( 3x-4 ) }}}
{{{ x^2 = 6x - 8 }}}
{{{ x^2 - 6x + 8 = 0 }}}
{{{ ( x - 4 )*( x - 2 ) = 0 }}} ( by inspection )
{{{ x = 4 }}}
{{{ x = 2 }}}
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Check the answers:
{{{ 2*log(4,x) - log(4, 3x - 4) = 1/2 }}}
{{{ 2*log(4,4) - log(4, 3*4 - 4) = 1/2 }}}
{{{ 2*1 - log( 4, 8 ) = 1/2 }}}
{{{ log( 4,8 ) = 2 - 1/2 }}}
{{{ log( 4,8 ) = 3/2 }}}
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This is true since
{{{ 4^(3/2) = 8 }}}
Square both sides
{{{ 4^3 = 64 }}}
{{{ 64 = 64 }}}
OK
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