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

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

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

Exponents

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