Certainly! Let's solve each of these logarithmic equations step-by-step:
### (a) [tex]\(\log_7 3 - \log_7 8 = \log_7 \square\)[/tex]
We use the property of logarithms that states:
[tex]\[
\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)
\][/tex]
Applying this property, we have:
[tex]\[
\log_7 3 - \log_7 8 = \log_7 \left(\frac{3}{8}\right)
\][/tex]
Thus, the missing value in the equation is:
[tex]\[
\square = \frac{3}{8}
\][/tex]
So, (a) [tex]\(\log_7 3 - \log_7 8 = \log_7 \left(\frac{3}{8}\right)\)[/tex].
### (b) [tex]\(\log_5 2 + \log_5 \square = \log_5 14\)[/tex]
We use the property of logarithms that states:
[tex]\[
\log_b a + \log_b c = \log_b (a \cdot c)
\][/tex]
Applying this property, we have:
[tex]\[
\log_5 2 + \log_5 \square = \log_5 (2 \cdot \square)
\][/tex]
Given that it equals [tex]\(\log_5 14\)[/tex], we can equate:
[tex]\[
2 \cdot \square = 14
\][/tex]
Solving for the missing value, we get:
[tex]\[
\square = \frac{14}{2} = 7
\][/tex]
So, (b) [tex]\(\log_5 2 + \log_5 7 = \log_5 14\)[/tex].
### (c) [tex]\(3 \log_3 2 = \log_3 \square\)[/tex]
We use the property of logarithms that states:
[tex]\[
n \log_b a = \log_b (a^n)
\][/tex]
Applying this property, we have:
[tex]\[
3 \log_3 2 = \log_3 (2^3)
\][/tex]
Thus, the missing value in the equation is:
[tex]\[
\square = 2^3 = 8
\][/tex]
So, (c) [tex]\(3 \log_3 2 = \log_3 8\)[/tex].
### Summary of Solutions:
(a) [tex]\(\log_7 3 - \log_7 8 = \log_7 \left(\frac{3}{8}\right)\)[/tex]
(b) [tex]\(\log_5 2 + \log_5 7 = \log_5 14\)[/tex]
(c) [tex]\(3 \log_3 2 = \log_3 8\)[/tex]
We have filled in the missing values to make the equations true.
