Question

Find a relation between $x,y$ when given that $x={\log }_{a}ab$ and $y={\log }_{b}ab$.

Answer 1

Given ${\log }_a^{ab}=x$ & ${\log }_a^{ab}=y$

Now, ${\log }_a^{ab}=Y_x-------\left(1\right)$ & ${\log }_{ab}^b=Y_y-----------\left(2\right)$

Adding $\left(1\right)$ & $\left(2\right)$ we get,

${\log }_{ab}^a+{\log }_{ab}^b=Y_x+Y_y$

or, ${\log }_{ab}^{ab}=(Y_x+Y_y)$

or, $1=Y_x+\frac{1}{y}$

or, $x+y=xy.$