Question

IF F(X)=LOG(X-1)/X SHOW THAT F(Y^2)=F(Y)+F(-Y)

Answer 1

Step-by-step explanation:

$\\\tt f(x) = log( \dfrac{x - 1}{x} )$

For

f(y^2)

$\\\tt f( {y}^{2} ) = log( \dfrac{ {y}^{2} - 1}{ {y}^{2} } )$

Let this be (i)

And f(y)

$\\\tt f(y) =log( \dfrac{y - 1}{y} )$

and f(-y)

$\\\tt f( - y) = log( \dfrac{ - y- 1}{ - y} ) \\\tt = log( \dfrac{y + 1}{y} )$

$\\\tt f(y) + f( - y) = log( \dfrac{y - 1}{y} ) + log( \dfrac{y + 1}{y} )$

Using sum rule,

$\\\tt log(x) + log(y) = log(xy)$

Sum will be:

$\\\tt f(y) + f( - y)= log( \dfrac{(y- 1)(y + 1)}{ {y}^{2} } ) \\\tt = log( \dfrac{ {y}^{2} - 1 }{ {y}^{2} } )$

Let this be (ii)

Since eq (i) == eq(ii)

Hence proved.