Chapter 3.2, Problem 121E

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 P3 of Theorem 3

$M=a^X$

Definition of logarithm

So, $M^k={\left(a^X\right)}^k$

if $u=v$, then $u^c=v^c$

$=a^{Xk}$

Power Rule for exponents

Thus, ${\log }_a{M}^k=Xk$

definition of logarithm

$=k\cdot {\log }_{a}M$. substitution and the commutative law for multiplication