Chapter 3.2, Problem 122E

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 P7 of Theorem 3

Let ${\log }_{b}M=R$

Then $b^R=M,$

definition of logarithm and ${\log }_a\left(b^R\right)={\log }_{a}M$. $\underset{\_}{\text{If }u=v,{\text{ then log}}_{c}u={\text{log}}_{c}v}$

Thus, $R\cdot {\log }_{a}b={\log }_{a}M.$

using Property P3 and $R=\frac{{\log }_{a}M}{{\log }_{a}b}$. $\underset{\_}{\text{If }a=b\text{ and }c\ne 0\text{ then }a/c=b/c}$

It follows that

${\log }_{b}M=\frac{{\log }_{a}M}{{\log }_a^b}$. substitution