Question

fill in the missing values to make the equations true.
\\(\log_{5} 8 - \log_{5} 11 = \log_{5} \square\\)
\\(\log_{4} 3 + \log_{4} \square = \log_{4} 15\\)
\\(\log_{7} 25 = \square \log_{7} 5\\)

Explanation:

Step1: Apply log subtraction rule

The logarithmic subtraction rule is $\log_b a - \log_b c = \log_b \frac{a}{c}$. For $\log_5 8 - \log_5 11$, we have $a = 8$, $c = 11$, so $\log_5 8 - \log_5 11 = \log_5 \frac{8}{11}$.

Step2: Apply log addition rule

The logarithmic addition rule is $\log_b a + \log_b c = \log_b (a \times c)$. For $\log_4 3 + \log_4 \square = \log_4 15$, let the missing value be $x$. Then $\log_4 (3 \times x) = \log_4 15$, so $3x = 15$, and $x = 5$.

Step3: Apply log power rule

The logarithmic power rule is $\log_b a^n = n\log_b a$. For $\log_7 25$, since $25 = 5^2$, we have $\log_7 25 = \log_7 5^2 = 2\log_7 5$.

Answer:

First equation: $\frac{8}{11}$; Second equation: $5$; Third equation: $2$