Which expression is equivalent to $\log _{3}\left(\frac{5}{7}\right)$ ? Select the correct answer below: $\log _{3}(5)-\log _{3}(7)$ $\frac{\log _{3}(5)}{\log _{3}(7)}$ $\log _{3}(7)-\log _{3}(5)$ $\frac{\log _{3}(7)}{\log _{3}(5)}$ $\log _{3}(-2)$ $\log _{3}(2)$
Solution
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: \( \boxed{\log _{3}(5)-\log _{3}(7)} \)
