How do you condense \[\log \left( r \right)-\log \left( t \right)-2\log \left( s \right)\]?

Hint: In order to find the solution of the given question that is to condense \[\log \left( r \right)-\log \left( t \right)-2\log \left( s \right)\] apply the formulas of the log rule that are \[\log \left( a \right)-\log \left( b \right)=\log \left( \dfrac{a}{b} \right)\] and \[a\log \left( b \right)=\log \left( {{a}^{b}} \right)\]. Then apply the fraction rule that is \[\dfrac{\dfrac{a}{b}}{c}=\dfrac{a}{b\cdot c}\] to get the condensed form of the given expression.

Complete step by step solution:
According to the question, given expression in the question is as follows:
\[\log \left( r \right)-\log \left( t \right)-2\log \left( s \right)\]
Now apply the log rule that is \[\log \left( a \right)-\log \left( b \right)=\log \left( \dfrac{a}{b} \right)\] in the above expression which means we can simplify the above expression by writing \[\log \left( r \right)-\log \left( t \right)\] as \[\log \left( \dfrac{r}{t} \right)\], so we can rewrite the above expression as:
\[\Rightarrow \log \left( \dfrac{r}{t} \right)-2\log \left( s \right)\]
After this apply the log rule that is \[a\log \left( b \right)=\log \left( {{a}^{b}} \right)\] in the above expression this signifies we can simplify the above expression more by writing \[2\log \left( s \right)\] as \[\log \left( {{s}^{2}} \right)\], so, we will have:
\[\Rightarrow \log \left( \dfrac{r}{t} \right)-\log \left( {{s}^{2}} \right)\]
Now we will apply the log rule that is \[\log \left( a \right)-\log \left( b \right)=\log \left( \dfrac{a}{b} \right)\] in the above expression this means we can simplify the above expression by writing \[\log \left( \dfrac{r}{t} \right)-\log \left( {{s}^{2}} \right)\] as \[\log \left( \dfrac{\dfrac{r}{t}}{{{s}^{2}}} \right)\], so we can rewrite the above expression as:
\[\Rightarrow \log \left( \dfrac{\dfrac{r}{t}}{{{s}^{2}}} \right)\]
After this to simplify the above expression, apply the fraction rule that is \[\dfrac{\dfrac{a}{b}}{c}=\dfrac{a}{b\cdot c}\] which signifies that we can further simplify above expression by writing \[\log \left( \dfrac{\dfrac{r}{t}}{{{s}^{2}}} \right)\] as \[\log \left( \dfrac{r}{t{{s}^{2}}} \right)\], we will get:
\[\Rightarrow \log \left( \dfrac{r}{t{{s}^{2}}} \right)\]
Therefore, the condensed form of the given expression \[\log \left( r \right)-\log \left( t \right)-2\log \left( s \right)\] is equal to \[\log \left( \dfrac{r}{t{{s}^{2}}} \right)\].

Note: Students make mistakes while applying the formulas of the log rule and end up using the formula \[\log \left( a \right)-\log \left( b \right)=\log \left( ab \right)\] which is completely incorrect and further leads to the wrong answer. Therefore, it’s important to remember that these are two different formulas of log rule and the correct formulas of those two this log rule are \[\log \left( a \right)-\log \left( b \right)=\log \left( \dfrac{a}{b} \right)\] and \[\log \left( ab \right)=\log \left( a \right)+\log \left( b \right)\].