I have an equation as below
$$2\log_2(1+xC)=\log_2(1+yC)$$
what is the relation between $x$ and $y$.
I mean I want to see the relation as $x=\alpha y$
Then what is the value of $\alpha$.
Do we have exact value or we must be satisfied with some approximate value.
If so, what is the exact/approximate value of $\alpha$?
On
Because of that "2" in front of the first logarithm, y is NOT a multiple of x. $$2\log_2(1+ Cx)= \log_2((1+ Cx)^2)= \log_2(1+ Cy).$$ so $$(1+ Cx)^2= 1+ 2Cx+ C^2x^2= 1+ Cy\\ Cy= C^2x^2+ 2Cx$$ and $$y= Cx^2+ 2x$$ $y$ is a quadratic function of $x$, not linear.
You can APPROXIMATE $y$ by a linear function near a given value of $x$. For example $y'= 2Cx+ C$. At $x= x_0$ that is $2Cx_0+ C$. We can approximate the function, near $x= x_0$ by $y= (2x_0+ C)(x- x_0)+ Cx_0^2+ 2x_0$
Use $2\log a=\log a^2$ than you have a very simple equation with gives you for $C\ne 0$ : $$y=x^2+2x$$