Chapter 3.2, Problem 120E

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 P2 of Theorem 3.

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

definition of logarithm

So $\frac{M}{N}=\frac{a^X}{a^Y}=a^{X-Y}$

Quotient Rule for exponents

Thus, ${\log }_a\frac{M}{N}=X-Y$

Definition of logarithm

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

substitution