Let's solve each part of the question step-by-step.
### Part (a)
We have the expression:
[tex]\[
\log_8 7 - \log_8 3
\][/tex]
Using the properties of logarithms, specifically, the difference rule:
[tex]\[
\log_b a - \log_b c = \log_b \left( \frac{a}{c} \right)
\][/tex]
So,
[tex]\[
\log_8 7 - \log_8 3 = \log_8 \left( \frac{7}{3} \right)
\][/tex]
Thus, the missing value is:
[tex]\[
\boxed{\frac{7}{3}}
\][/tex]
### Part (b)
We are given:
[tex]\[
\log_4 9 + \log_4 \square = \log_4 45
\][/tex]
Using the properties of logarithms, specifically, the addition rule:
[tex]\[
\log_b a + \log_b c = \log_b (a \cdot c)
\][/tex]
So,
[tex]\[
\log_4 9 + \log_4 x = \log_4 (9 \cdot x)
\][/tex]
For the equation to be true,
[tex]\[
9 \cdot x = 45
\][/tex]
Solving for [tex]\(x\)[/tex],
[tex]\[
x = \frac{45}{9} = 5
\][/tex]
Thus, the missing value is:
[tex]\[
\boxed{5}
\][/tex]
### Part (c)
We are given:
[tex]\[
-3 \log_3 2 = \log_3 \square
\][/tex]
Using the properties of logarithms, specifically, the power rule:
[tex]\[
a \log_b c = \log_b (c^a)
\][/tex]
So,
[tex]\[
-3 \log_3 2 = \log_3 (2^{-3})
\][/tex]
Simplifying the expression:
[tex]\[
2^{-3} = \frac{1}{2^3} = \frac{1}{8}
\][/tex]
Thus, the missing value is:
[tex]\[
\boxed{\frac{1}{8}}
\][/tex]
Therefore, the filled-in equations are:
(a) [tex]\(\log_8 7 - \log_8 3 = \log_8 \boxed{\frac{7}{3}}\)[/tex]
(b) [tex]\(\log_4 9 + \log_4 \boxed{5} = \log_4 45\)[/tex]
(c) [tex]\(-3 \log_3 2 = \log_3 \boxed{\frac{1}{8}}\)[/tex]
