Answer :
Sure, let's solve each part step by step.
### Part (a)
[tex]\[ \log_8 3 + \log_8 7 = \log_8 (\boxed{\phantom{}}) \][/tex]
Using the property of logarithms that states:
[tex]\[ \log_b(a) + \log_b(c) = \log_b(a \cdot c) \][/tex]
we combine the two logarithms:
[tex]\[ \log_8 3 + \log_8 7 = \log_8 (3 \cdot 7) \][/tex]
[tex]\[ \log_8 3 + \log_8 7 = \log_8 21 \][/tex]
So, the missing value is:
[tex]\[ \boxed{21} \][/tex]
### Part (b)
[tex]\[ \log_4 5 - \log_4 \boxed{\phantom{}} = \log_4 \frac{5}{7} \][/tex]
Using the property of logarithms that states:
[tex]\[ \log_b(a) - \log_b(c) = \log_b \left(\frac{a}{c}\right) \][/tex]
we can set up the equation:
[tex]\[ \log_4 5 - \log_4 7 = \log_4 \left(\frac{5}{7}\right) \][/tex]
So, the missing value is:
[tex]\[ \boxed{7} \][/tex]
### Part (c)
[tex]\[ \log_5 \frac{1}{9} = -2 \log_5 \boxed{\phantom{}} \][/tex]
Using the property of logarithms that states:
[tex]\[ \log_b \left(a^c\right) = c \log_b(a) \][/tex]
we rewrite the equation:
[tex]\[ \log_5 \frac{1}{9} = -2 \log_5 3 \][/tex]
Recognize that:
[tex]\[ 3^2 = 9 \][/tex]
and
[tex]\[ \frac{1}{9} = 3^{-2} \][/tex]
Thus,
[tex]\[ \log_5 \frac{1}{9} = \log_5 (3^{-2}) \][/tex]
which simplifies by the logarithmic power rule:
[tex]\[ \log_5 (3^{-2}) = -2 \log_5 3 \][/tex]
So, the missing value is:
[tex]\[ \boxed{3} \][/tex]
To summarize, the missing values are:
(a) [tex]\(\boxed{21}\)[/tex]
(b) [tex]\(\boxed{7}\)[/tex]
(c) [tex]\(\boxed{3}\)[/tex]
### Part (a)
[tex]\[ \log_8 3 + \log_8 7 = \log_8 (\boxed{\phantom{}}) \][/tex]
Using the property of logarithms that states:
[tex]\[ \log_b(a) + \log_b(c) = \log_b(a \cdot c) \][/tex]
we combine the two logarithms:
[tex]\[ \log_8 3 + \log_8 7 = \log_8 (3 \cdot 7) \][/tex]
[tex]\[ \log_8 3 + \log_8 7 = \log_8 21 \][/tex]
So, the missing value is:
[tex]\[ \boxed{21} \][/tex]
### Part (b)
[tex]\[ \log_4 5 - \log_4 \boxed{\phantom{}} = \log_4 \frac{5}{7} \][/tex]
Using the property of logarithms that states:
[tex]\[ \log_b(a) - \log_b(c) = \log_b \left(\frac{a}{c}\right) \][/tex]
we can set up the equation:
[tex]\[ \log_4 5 - \log_4 7 = \log_4 \left(\frac{5}{7}\right) \][/tex]
So, the missing value is:
[tex]\[ \boxed{7} \][/tex]
### Part (c)
[tex]\[ \log_5 \frac{1}{9} = -2 \log_5 \boxed{\phantom{}} \][/tex]
Using the property of logarithms that states:
[tex]\[ \log_b \left(a^c\right) = c \log_b(a) \][/tex]
we rewrite the equation:
[tex]\[ \log_5 \frac{1}{9} = -2 \log_5 3 \][/tex]
Recognize that:
[tex]\[ 3^2 = 9 \][/tex]
and
[tex]\[ \frac{1}{9} = 3^{-2} \][/tex]
Thus,
[tex]\[ \log_5 \frac{1}{9} = \log_5 (3^{-2}) \][/tex]
which simplifies by the logarithmic power rule:
[tex]\[ \log_5 (3^{-2}) = -2 \log_5 3 \][/tex]
So, the missing value is:
[tex]\[ \boxed{3} \][/tex]
To summarize, the missing values are:
(a) [tex]\(\boxed{21}\)[/tex]
(b) [tex]\(\boxed{7}\)[/tex]
(c) [tex]\(\boxed{3}\)[/tex]
