Question

prove the quotient rule using a power m/ a power n = a power m-n​

Answer 1

Given

$\frac{a^{m} }{a^{n} }$

To proof

$\frac{a^{m} }{a^{n} }\ = a^{m-n}$

Solution

By using the quotient property of log that can be written as

$log_{b}\ (\frac{a}{b}\ ) \ = \ log_{b}\ ( a)\ - \ log_{b} (b)$

$log_{b}\ (\frac{a}{b}\ ) \ = \ log_{b} \ a^{m-n}$

Apply this property into the solution

Apply the log with the base b in the left side of the given  question

$log_{b}\ (\frac{a}{b}\ )$

this can be written as

$log_{b}\ (\frac{a}{b}\ ) \ = \ log_{b}\ ( a)\ - \ log_{b} (b)$

$log_{b}\ (\frac{a}{b}\ ) \ = \ log_{b} \ a^{m-n}$

This can be written as

$a^{m-n}$

that is equal to the RHS

Hence proved