Product Law of Logarithms

The logarithm of a product is equal to the sum of the logarithms of the factors:

Product Law

${\log }_b(mn)={\log }_{b}m+{\log }_{b}n$

Proof

$b^x=m$ $x={\log }_{b}m$ $b^y=n$ $y={\log }_{b}n$ Make them products $b^y\ast b^x=mn$ $b^{x+y}=mn$ $x+y={\log }_b(mn)$ ${\log }_{b}m+lo{b}_{b}n={\log }_b(mn)$

Power Law & Product/Quotient law

Say we had a question like: $3\log (x)=\log (2x)$ we must use product law to bring earth back to heaven $\log (x^3)=\log (2x)$ Then solve like normal $\log (x^2)-\log (2x)=0$ $\log (\frac{x^2}{2x})=0$ $\log (\frac{x}{2})=0$ $\frac{x}{2}=1$ $x=2$