Sum of Logarithms/Natural Logarithm/Proof 1

Proof
Let $y \in \R_{>0}$ be fixed.

Consider the function:
 * $\map f x = \ln x y - \ln x$

From the definition of the natural logarithm, the Fundamental Theorem of Calculus and the Chain Rule for Derivatives:


 * $\forall x > 0: \map {f'} x = \dfrac 1 {x y} y - \dfrac 1 x = \dfrac 1 x - \dfrac 1 x = 0$

Thus from Zero Derivative implies Constant Function, $f$ is constant:
 * $\forall x > 0: \ln x y - \ln x = c$

To determine the value of $c$, put $x = 1$.

From Logarithm of 1 is 0:
 * $\ln 1 = 0$

Thus:
 * $c = \ln y - \ln 1 = \ln y$

and hence the result.