Question 1029314
{{{ log( 4,sqrt( x+3 )) = 1/4 + log( 4,sqrt( 2x-1 ) ) }}}
{{{ log( 4,sqrt( x+3 )) -  log( 4,sqrt( 2x-1 ) ) =  1/4  }}}
{{{ log( 4, sqrt( (x+3)/(2x-1) )) = 1/4 }}}
{{{ (1/2)*log( 4,  (x+3)/(2x-1) ) = 1/4 }}}
{{{ log( 4,  (x+3)/(2x-1) ) = 1/2 }}}
{{{ log( 4,  (x+3)/(2x-1) ) = log( 4, 2 ) }}}
{{{ ( x + 3 ) / ( 2x - 1 ) = 2 }}}
{{{ x + 3 = 2*( 2x - 1 ) }}}
{{{ x + 3 = 4x - 2 }}}
{{{ 3x = 5 }}}
{{{ x = 5/3 }}}
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check answer:
{{{ log( 4,sqrt( x+3 )) = 1/4 + log( 4,sqrt( 2x-1 ) ) }}}
{{{ log( 4,sqrt( 5/3+9/3 )) = 1/4 + log( 4,sqrt( 2*(5/3) -3/3 ) ) }}}
{{{ log( 4,sqrt( 14/3 )) = 1/4 + log( 4,sqrt( 7/3 ) ) }}}
{{{ (1/2)*log( 4, 14/3 ) - (1/2)*log( 4, 7/3 ) = 1/4 }}}
{{{ (1/2)*( log( 4, 14/3 ) - log( 4, 7/3 ) ) = 1/4 }}}
{{{ log( 4, 14/3 ) - log( 4, 7/3 ) = 1/2 }}}
{{{ log( 4, 14/3 ) - log( 4, 7/3 ) = log( 4,2 ) }}}
{{{ log( 4, ( ( 14/3)/( 7/3) ) ) = log( 4,2 ) }}}
{{{ 14/7 = 2 }}}
{{{ 2= 2 }}}
OK
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It is also true that
{{{ log( 4,  (x+3)/(2x-1) ) = 1/2 }}}
{{{ log( 4,  (x+3)/(2x-1) ) = log( 4, -2 ) }}}
You can figure that out ( if there is an answer )
Hope this helps