Logarithm Formula

Logarithms are the opposite phenomena of exponential like subtraction is the inverse of addition process, and division is the opposite phenomena of multiplication. Logs “undo” exponentials.

Basic Logarithm Formulas

\[\large \log _{b} (xy) = \log _{b}(x) + \log _{b}(y)\]

\[\large \log _{b}\left ( \frac{x}{y} \right ) = \log _{b}(x) – \log _{b}(y)\]

\[\large \log_{b}(x^{d})= d \log_{b}(x)\]

\[\large \log_{b}(\sqrt[y]{x})= \frac{\log_{b}(x)}{y}\]

\[\large c\log_{b}(x)+d\log_{b}(y)= \log_{b}(x^{c}y^{d})\]

Changing the Base

\[\large \log_{b}a = \frac{\log_{d}(a)}{\log_{d}(b)}\]

Addition & Subtraction

\[\large \log_{b} (a+c) = \log_{b}a + \log_{b}\left ( 1 + \frac{c}{a} \right )\]

\[\large \log_{b} (a-c) = \log_{b}a + \log_{b}\left ( 1 – \frac{c}{a} \right )\]


\[\large x^{\frac{\log(\log(x))}{\log(x)}} \; = \; \log(x)\]


More topics in Logarithm Formula
Natural Log Formula Change of Base Formula
Exponential Growth Formula

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