Question

fill in the missing values to make the equations true.
\\(\log_{3}7 - \log_{3}2 = \log_{3}\square\\)
\\(\log_{9}4 + \log_{9}\square = \log_{9}20\\)
\\(4\log_{6}2 = \log_{6}\square\\)

Explanation:

Step1: Apply log subtraction rule

The logarithmic subtraction rule is $\log_b a - \log_b c = \log_b \frac{a}{c}$. For $\log_3 7 - \log_3 2$, we have $a = 7$, $c = 2$. So $\log_3 7 - \log_3 2 = \log_3 \frac{7}{2}$.

Step2: Apply log addition rule

The logarithmic addition rule is $\log_b a + \log_b c = \log_b (a \times c)$. For $\log_9 4 + \log_9 \square = \log_9 20$, let the missing value be $x$. Then $\log_9 (4 \times x) = \log_9 20$, so $4x = 20$, $x = 5$.

Step3: Apply log power rule

The logarithmic power rule is $n\log_b a = \log_b a^n$. For $4\log_6 2$, we have $n = 4$, $a = 2$. So $4\log_6 2 = \log_6 2^4 = \log_6 16$.

Answer:

First equation: $\frac{7}{2}$; Second equation: $5$; Third equation: $16$