Sum of Logarithms

= Natural Logarithms =

Theorem
Let $$x, y \in \mathbb{R}$$ be strictly positive real numbers.

Then $$\ln x + \ln y = \ln x y$$.

Proof
Let $$y \in \mathbb{R}, y > 0$$ be fixed.

Consider the function $$f \left({x}\right) = \ln xy - \ln x$$.

Then from the definition of the natural logarithm, the Fundamental Theorem of Calculus and the Chain Rule:

$$\forall x > 0: f^{\prime} \left({x}\right) = \frac 1 {xy} y - \frac 1 x = \frac 1 x - \frac 1 x = 0$$.

Thus from Zero Derivative means Constant Function, $$f$$ is constant: $$\forall x > 0: \ln xy - \ln x = c$$.

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

From Basic Properties of Natural Logarithm, $$\ln 1 = 0$$.

Thus $$c = \ln y - \ln 1 = \ln y$$.