Using properties of determinant show that`|(1,log_x y,log_x z),(log_y x,1,log_y z),(log_z x,log_z y,1)|=0`

Select and write the correct answer from the given alternatives in each of the following:If x>0 andx!=1,y>0 and y!=1,z>0 and z!=1, then the value of |(1,log_x (y),log_x (z)),(log_y (x),1,log_y (z)),(log_z (x),log_z (y),1)| is

By using properties of determinant prove that |(x+y,y+z,z+x),(z,x,y),(1,1,1)|=0

Without expanding the determinants, show that |(1,x,x^2),(1,y,y^2),(1,z,z^2)|=(y-x)(z-y)(z-x)

If x^y = e^(x-y) , then show that dy/dx = frac{log x}{(1+ log x)^2}

If e^y = y^x , then show that dy/dx = frac{(log y)^2}{log y -1}