# 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

$\large \log _{b} (1) = 0; \; because \; b^{0}=1; \; b> 0$

$\large \log _{b} (b) = 1; \; because \; b^{1}=b$

#### 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

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

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

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

#### 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
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