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
\[\large x^{\frac{\log(\log(x))}{\log(x)}} \; = \; \log(x)\]
| More topics in Logarithm Formula | |
| Natural Log Formula | Change of Base Formula |
| Exponential Growth Formula | |