Sum of Logarithms

Theorem


This page has been identified as a candidate for refactoring.
In particular: revisit nesting
Until this has been finished, please leave {{Refactor}} in the code.

New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only.

Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


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


Sources

  • 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $5$: Eternal Triangles: Logarithms
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.