If log(x-y/2)=1/2(logx+logy-logz), then find the value of x/y+y/x

Asked by devulapallyvaishnavi | 29th Oct, 2015, 08:44: PM

Expert Answer:

C o n s i d e r space t h e space g i v e n space e q u a t i o n comma log open parentheses fraction numerator x minus y over denominator 2 end fraction close parentheses equals 1 half open parentheses log x plus log y minus log z close parentheses T a k i n g space e x p o n e n t space o n space b o t h space t h e space s i d e s comma space w e space h a v e comma e to the power of l o g open parentheses fraction numerator x minus y over denominator 2 end fraction close parentheses end exponent equals e to the power of 1 half open parentheses l o g x plus l o g y minus l o g z close parentheses space space space space space end exponent space space space space space rightwards double arrow open parentheses fraction numerator x minus y over denominator 2 end fraction close parentheses equals e to the power of fraction numerator l o g x over denominator 2 end fraction end exponent cross times e to the power of fraction numerator l o g y over denominator 2 end fraction end exponent cross times e to the power of negative fraction numerator l o g z over denominator 2 end fraction end exponent space space space space space space space open square brackets because e to the power of l o g a end exponent equals a close square brackets rightwards double arrow open parentheses fraction numerator x minus y over denominator 2 end fraction close parentheses equals fraction numerator e to the power of fraction numerator log x over denominator 2 end fraction end exponent cross times e to the power of fraction numerator log y over denominator 2 end fraction end exponent over denominator e to the power of negative fraction numerator l o g z over denominator 2 end fraction end exponent end fraction rightwards double arrow open parentheses fraction numerator x minus y over denominator 2 end fraction close parentheses equals fraction numerator open parentheses e to the power of log x end exponent close parentheses to the power of begin display style 1 half end style end exponent cross times open parentheses e to the power of l o g y end exponent close parentheses to the power of 1 half end exponent over denominator open parentheses e to the power of l o g z end exponent close parentheses to the power of 1 half end exponent end fraction...... open square brackets because k log a equals log a to the power of k close square brackets rightwards double arrow open parentheses fraction numerator x minus y over denominator 2 end fraction close parentheses equals fraction numerator x to the power of begin display style 1 half end style end exponent cross times y to the power of 1 half end exponent over denominator z to the power of 1 half end exponent end fraction...... open square brackets because e to the power of l o g a end exponent equals a close square brackets S q u a r i n g space b o t h space t h e space s i d e s comma space w e space h a v e comma rightwards double arrow open parentheses fraction numerator x minus y over denominator 2 end fraction close parentheses squared equals fraction numerator x y over denominator z end fraction rightwards double arrow open parentheses fraction numerator x squared plus y squared minus 2 x y over denominator 4 end fraction close parentheses equals fraction numerator x y over denominator z end fraction rightwards double arrow open parentheses fraction numerator x squared plus y squared minus 2 x y over denominator x y end fraction close parentheses equals 4 over z rightwards double arrow open parentheses fraction numerator x squared plus y squared over denominator x y end fraction minus fraction numerator 2 x y over denominator x y end fraction close parentheses equals 4 over z rightwards double arrow fraction numerator x squared plus y squared over denominator x y end fraction minus 2 equals 4 over z rightwards double arrow fraction numerator x squared plus y squared over denominator x y end fraction equals 4 over z plus 2 rightwards double arrow x over y plus y over x equals fraction numerator 4 plus 2 z over denominator z end fraction

Answered by Vimala Ramamurthy | 30th Oct, 2015, 10:32: AM

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