Sum of Logarithms/Natural Logarithm/Proof 5

Theorem

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


Then:

$\ln x + \ln y = \map \ln {x y}$

where $\ln$ denotes the natural logarithm.


Proof

\(\ds x y\) \(=\) \(\ds x \times y\)
\(\ds \leadsto \ \ \) \(\ds e^{\map \ln {xy} }\) \(=\) \(\ds e^{\ln x} \times e^{\ln y}\) Exponential of Natural Logarithm
\(\ds \leadsto \ \ \) \(\ds e^{\map \ln {xy} }\) \(=\) \(\ds e^{\ln x + \ln y}\) Product of Powers
\(\ds \leadsto \ \ \) \(\ds \map \ln {xy}\) \(=\) \(\ds \ln x + \ln y\)

$\blacksquare$


Sources

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