4.2.1 Laws of Logarithms

$$\boxed{\begin{array}{l}\text{ Law 1:}\\ {\text{ log}}_{a}xy={\text{log}}_{a}x+{\text{log}}_{a}y\\ \\ \text{ Example:}\\ {\log }_{5}25x={\text{log}}_{5}25+{\text{log}}_{5}x\\ \\ \text{ Beware!!}\\ {\text{ log}}_{a}x+{\text{log}}_{a}y\ne {\text{log}}_a\left(x+y\right)\end{array}}$$
$$\boxed{\begin{array}{l}\text{ Law 2:}\\ {\text{ log}}_a\left(\frac{x}{y}\right)={\text{log}}_{a}x-{\text{log}}_{a}y\\ \\ \text{ Example:}\\ {\log }_5\frac{x}{25}={\text{log}}_{5}x-{\text{log}}_{5}25\\ \\ \text{ Beware!!}\\ {\text{ log}}_a\frac{x}{y}\ne \frac{{\text{log}}_{a}x}{{\text{log}}_{a}y}\end{array}}$$
$$\boxed{\begin{array}{l}\text{ Law 3:}\\ {\text{ log}}_a{x}^m=m{\text{log}}_{a}x\\ \\ \text{ Example:}\\ {\log }_5{y}^5=5{\text{log}}_{5}y\\ \\ \text{ Beware!!}\\ {\left({\text{log}}_{a}x\right)}^2\ne 2{\text{log}}_{a}x\end{array}}$$

Example 1

Express the following in term of ${\log }_{a}x$   and ${\log }_{a}y$  .
(a) ${\log }_{a}3x$
(b) ${\log }_a\frac{x}{5}$
(c) ${\log }_a{y}^5$
(d) ${\log }_{a}x{y}^3$
(e) ${\log }_a\frac{x^2}{\sqrt{y}}$
(f) ${\log }_a\sqrt{\frac{y}{a^2{x}^3}}$

Law of Logarithms