Chapter 3.2, Problem 119E

To prove Proprieties P1, P2, P3, and P7 of Theorem 3, let $X={\log }_{a}M$ and $Y={\log }_{a}N,$

and give reasons for the steps listed in Exercises 119 – 122.

Proof of P1 of Theorem 3

$M=a^X$ and $N=a^Y,$

definition of logarithm

So $MN=a^X\cdot a^Y=a^X+y$

Product Rule for exponents

Thus, ${\log }_a\left(MN\right)=X+Y$

definition of logarithm

$={\log }_{a}M+{\log }_{a}N$. substitution