log identity of division with two variables

Question

I'm having a hard time figuring out the log of a fraction with two variables.

For instance, $$f(x,y) = \frac{x}{x+y}$$ and if I took $$\log(f(x,y))$$ what would it be?

I know that $$\log(x/y) = \log(x) - \log(y)$$ so does that mean, my equation will deconstruct into $$\log(\frac{x}{x+y}) = \log(x) - \log(x+y)$$

Answer 1

Yes of course we can write

$$\log\left(\frac{x}{x+y}\right) = \log(x) - \log(x+y)$$

with the limitations

• $\frac{x}{x+y}>0$
• $x>0$
• $x+y>0$

that is $x>0$ and $x+y>0$.

Answer 2

You also can have a simpler constraint, rewriting $$\log f(x,y)=\log\biggl(\frac 1{1+\frac yx}\biggr)=-\log\Bigl(1+\frac yx\Bigr),$$ which requires $x\ne 0$, $\;\dfrac yx>-1$.