5.2a Laws of Logarithms

Posted on April 21, 2020 by user

5.2a Laws of Logarithms

$\begin{array}{l} Law 1: \\ {log}_{a} x y = {log}_{a} x + {log}_{a} y \\ Example: \\ \log_{5} 25 x = {log}_{5} 25 + {log}_{5} x \\ Beware!! \\ {log}_{a} x + {log}_{a} y \neq {log}_{a} (x + y) \end{array}$

$\begin{array}{l} Law 2: \\ {log}_{a} (\frac{x}{y}) = {log}_{a} x - {log}_{a} y \\ Example: \\ \log_{5} \frac{x}{25} = {log}_{5} x - {log}_{5} 25 \\ Beware!! \\ {log}_{a} \frac{x}{y} \neq \frac{{log}_{a} x}{{log}_{a} y} \end{array}$

$\begin{array}{l} Law 3: \\ {log}_{a} x^{m} = m {log}_{a} x \\ Example: \\ \log_{5} y^{5} = 5 {log}_{5} y \\ Beware!! \\ {({log}_{a} x)}^{2} \neq 2 {log}_{a} x \end{array}$