Sum of Logarithms

Theorem
Let $$x, y, b \in \R$$ be strictly positive real numbers.

Then:
 * $$\log_b x + \log_b y = \log_b \left({x y}\right)$$

where $$\log_b$$ denotes the logarithm to base $b$.

Proof for Natural Logarithm
First we demonstrate the result for the natural logarithm, i.e. when $$b$$ is Euler's number $$e$$.

Let $$y \in \R, y > 0$$ be fixed.

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

where we use the notation $$\ln$$ to mean $$\log_e$$.

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 $$c = \ln y - \ln 1 = \ln y$$, and hence the result.

Proof for General Logarithm
Next we expand the proof for logarithms to the general base $b$.

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