Sum of Logarithms

Theorem

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Natural Logarithm

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.

Complex Logarithm

Let $x, y \in \C$ where $x = r_1 e^{i \theta_1}$ and $y = r_2 e^{i \theta_2}$

Where:

$r_1$ and $r_2$ are both (strictly) positive real numbers.

Then:

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

where $\ln$ denotes the complex natural logarithm.

General Logarithm

Let $x, y, b \in \R$ be strictly positive real numbers such that $b > 1$.

Then:

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

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

Also see

• Logarithm of Infinite Product of Real Numbers

Sources

• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $5$: Eternal Triangles: Logarithms

• For a video presentation of the contents of this page, visit the Khan Academy.