To prove (1) directly from the laws of exponents, let $$x=\log _{b} A \text { and } y=\log _{b} B$$ by rewriting each logarithm in exponential format, show that $$A=b^{x} \text { and } B=b^{y}$$ then $$A B=b^{x} b^{y}$$ next show $$A B=b^{x+y}$$ and finally, $$\log _{b} A B=x+y=\log _{b} A+\log _{b} B$$