Question

which of the following is a correct way to rewrite log₃ 20 / log₃ 5? select all that apply.
a. log₃ 4
b. log₃ 20 / log₃ 5
c. log₃(20 - 5)
d. log₃ 20 - log₃ 5

Explanation:

Step1: Recall the quotient - rule of logarithms

The quotient - rule states that $\log_aM-\log_aN = \log_a\frac{M}{N}$. For the expression $\log_3{20}-\log_3{5}$, where $a = 3$, $M = 20$ and $N = 5$, we have $\log_3{20}-\log_3{5}=\log_3\frac{20}{5}=\log_3{4}$.

Step2: Analyze each option

• Option A: $\log_3{4}$ is correct as we derived above.
• Option B: $\frac{\log_3{20}}{\log_3{5}}$ is incorrect. By the change - of - base formula, $\frac{\log_3{20}}{\log_3{5}}=\log_5{20}

eq\log_3{20}-\log_3{5}$.

• Option C: $\log_3{(20 - 5)}=\log_3{15}

eq\log_3{20}-\log_3{5}$.

• Option D: $\log_3{20}-\log_3{5}$ is the original expression, which is equivalent to $\log_3{4}$ after applying the quotient rule.

Answer:

A. $\log_3{4}$, D. $\log_3{20}-\log_3{5}$