Sure, let's go through each equation step-by-step and identify the missing values:
(a) [tex]\(\log_9 10 - \log_9 7 = \log_9 ?\)[/tex]
We can use the properties of logarithms here. Specifically, we use the property that states:
[tex]\[
\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)
\][/tex]
Putting the given values into this form:
[tex]\[
\log_9 10 - \log_9 7 = \log_9 \left(\frac{10}{7}\right)
\][/tex]
Thus, the missing value is:
[tex]\[
\boxed{1.4285714285714286}
\][/tex]
(b) [tex]\(\log_7 3 + \log_7 \square = \log_7 33\)[/tex]
Again, we use another property of logarithms:
[tex]\[
\log_b A + \log_b B = \log_b (A \cdot B)
\][/tex]
Setting this up with the given values:
[tex]\[
\log_7 3 + \log_7 ? = \log_7 (3 \cdot ?)
\][/tex]
Since we know:
[tex]\[
\log_7 3 + \log_7 ? = \log_7 33
\][/tex]
We can infer:
[tex]\[
3 \cdot ? = 33 \quad \Rightarrow \quad ? = \frac{33}{3} = 11
\][/tex]
So, the missing value is:
[tex]\[
\boxed{11}
\][/tex]
(c) [tex]\(2 \log_9 2 = \log_9 \square\)[/tex]
Here, we can use the property that states:
[tex]\[
k \log_b A = \log_b (A^k)
\][/tex]
Using [tex]\(k=2\)[/tex] and [tex]\(A=2\)[/tex], we get:
[tex]\[
2 \log_9 2 = \log_9 (2^2)
\][/tex]
So:
[tex]\[
2^2 = 4
\][/tex]
Thus, the missing value is:
[tex]\[
\boxed{4}
\][/tex]
In summary, we've filled in the missing values as follows:
(a) [tex]\(\log_9 10 - \log_9 7 = \log_9 \left(\frac{10}{7}\right) \implies \boxed{1.4285714285714286}\)[/tex]
(b) [tex]\(\log_7 3 + \log_7 11 = \log_7 33 \implies \boxed{11}\)[/tex]
(c) [tex]\(2 \log_9 2 = \log_9 4 \implies \boxed{4}\)[/tex]
