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As, log of a to the base b = log a/log b\r
\n" ); document.write( "\n" ); document.write( "So, log of 9 to the base square root of x = log 9/log (square root of x)
\n" ); document.write( "=> log 9/log(x^1/2) = > \r
\n" ); document.write( "\n" ); document.write( "As, log(a^b) = bloga
\n" ); document.write( "=> log 9/log(x^1/2) = log (3^2)/log(x^1/2) = 2log3/((1/2)logx) = 4log3/logx...(1)\r
\n" ); document.write( "\n" ); document.write( "log of 3 to the base x = log 3/logx...(2)\r
\n" ); document.write( "\n" ); document.write( "log ofx to the base square root of 3 = log x/log (3^1/2) = log x/ ((1/2)log3) = 2logx/log3...(3)\r
\n" ); document.write( "\n" ); document.write( "Now according to the quesiton and using the equations (1)(2) and (3)
\n" ); document.write( "we get
\n" ); document.write( "4log3/logx - log3/logx = 9 + 6(2logx/log3)
\n" ); document.write( "=> 3log3/logx = 9 + 12logx/log3
\n" ); document.write( "=> 3log3/logx = (9log3 + 12logx)/log3
\n" ); document.write( "=> 3log3*log3 = 9log3*logx + 12logx*logx\r
\n" ); document.write( "\n" ); document.write( "Let logx be X and log3 be Y
\n" ); document.write( "=> 3Y*Y = 9YX + 12 X*X
\n" ); document.write( "=> 12X^2 + 9XY - 3Y^2 = 0
\n" ); document.write( "=> 4X^2 + 3XY - Y^2 = 0
\n" ); document.write( "=> 4X^2 + 4XY-XY -Y^2 = 0
\n" ); document.write( "=> 4X(X+Y)-Y(X+Y)=0
\n" ); document.write( "=> (4X-Y)(X+Y)=0\r
\n" ); document.write( "\n" ); document.write( "According to the zero product rule
\n" ); document.write( "4X-Y =0 and X+Y = 0
\n" ); document.write( "=> X = Y/4 and X = -Y
\n" ); document.write( "as Y = log 3
\n" ); document.write( "and X = logx
\n" ); document.write( "So, X= Y/4
\n" ); document.write( "=> logx = (log 3)/4 = log (3^1/4)
\n" ); document.write( "=> x = 3^1/4\r
\n" ); document.write( "\n" ); document.write( "X = -Y
\n" ); document.write( "=> logx = -log 3 = log 3^-1 = log (1/3)
\n" ); document.write( "so x = 1/3 \n" ); document.write( "