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.

Trivial Identities

\(\large \log _{b} (1) = 0\) Since, \( b^{0}=1; \; b> 0\)

\(\large \log _{b} (b) = 1\) Since, \(b^{1}=b\)

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

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

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

Exponents

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