Which expression is equivalent to ${\log }_3\left(\frac{5}{7}\right)$ ?

Select the correct answer below:
${\log }_3(5)-{\log }_3(7)$
$\frac{{\log }_3(5)}{{\log }_3(7)}$
${\log }_3(7)-{\log }_3(5)$
$\frac{{\log }_3(7)}{{\log }_3(5)}$
${\log }_3(-2)$
${\log }_3(2)$

Step 1 ：The question is asking for an equivalent expression to ${\log }_3\left(\frac{5}{7}\right)$.

Step 2 ：From the properties of logarithms, we know that ${\log }_b(a/c)={\log }_b(a)-{\log }_b(c)$.

Step 3 ：Therefore, the equivalent expression should be ${\log }_3(5)-{\log }_3(7)$.

Step 4 ：We can verify this by calculating both expressions and comparing the results. If they are equal, then the expressions are equivalent.

Step 5 ：The results are not exactly equal due to the precision of floating point arithmetic. However, they are very close to each other.

Step 6 ：This suggests that the expressions are indeed equivalent, but the slight difference in the results is due to the limitations of numerical precision in the calculations.

Step 7 ：Therefore, I can conclude that the equivalent expression to ${\log }_3\left(\frac{5}{7}\right)$ is ${\log }_3(5)-{\log }_3(7)$.

Step 8 ：Final Answer: $\boxed{{\displaystyle {\log }_3(5)-{\log }_3(7)}}$