Let's analyze and simplify each of the given logarithmic expressions step-by-step.
### (i) [tex]\(\log ( a + b ) + \log ( a - b ) - \log \left(a^2 - b^2\right)\)[/tex]
We start by using the properties of logarithms:
1. [tex]\(\log(a) + \log(b) = \log(a \cdot b)\)[/tex]
2. [tex]\(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\)[/tex]
Applying these properties:
[tex]\[
\log(a + b) + \log(a - b) = \log((a + b)(a - b))
\][/tex]
Now we have:
[tex]\[
\log((a + b)(a - b)) - \log(a^2 - b^2)
\][/tex]
We know that [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex], so we substitute this:
[tex]\[
\log(a^2 - b^2) - \log(a^2 - b^2) = \log\left(\frac{a^2 - b^2}{a^2 - b^2}\right) = \log(1)
\][/tex]
Finally, we know that [tex]\(\log(1) = 0\)[/tex], hence, the result is:
[tex]\[
0
\][/tex]
### (ii) [tex]\(\log 2 + 2 \log 5 - \log 3 - 2 \log 7\)[/tex]
We can use the properties of logarithms to combine and simplify the expression:
1. [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex]
2. [tex]\(\log(a) + \log(b) = \log(a \cdot b)\)[/tex]
3. [tex]\(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\)[/tex]
First, simplify [tex]\(2 \log 5\)[/tex] and [tex]\(2 \log 7\)[/tex]:
[tex]\[
2 \log 5 = \log(5^2) = \log(25)
\][/tex]
[tex]\[
2 \log 7 = \log(7^2) = \log(49)
\][/tex]
Now, the expression becomes:
[tex]\[
\log 2 + \log 25 - \log 3 - \log 49
\][/tex]
Combine the logarithms:
[tex]\[
\log 2 + \log 25 = \log(2 \cdot 25) = \log(50)
\][/tex]
[tex]\[
\log 3 + \log 49 = \log(3 \cdot 49) = \log(147)
\][/tex]
Now, we have:
[tex]\[
\log(50) - \log(147) = \log\left(\frac{50}{147}\right)
\][/tex]
Thus, the simplified result is:
[tex]\[
\log\left(\frac{50}{147}\right)
\][/tex]
### (iii) [tex]\(\log _7 8 \times \log _8 7\)[/tex]
We use the change of base formula:
[tex]\[ \log_b(a) = \frac{1}{\log_a(b)} \][/tex]
Thus,
[tex]\[
\log_7(8) \times \log_8(7) = 1
\][/tex]
So the result is:
[tex]\[
1
\][/tex]
### (iv) [tex]\(\frac{1}{2} \log 9 + \frac{1}{4} \log 81 + 2 \log 6 - \log 12\)[/tex]
Let's simplify each term using the properties of logarithms. Particularly, [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex]:
[tex]\[
\frac{1}{2} \log 9 = \log(9^{1/2}) = \log(\sqrt{9}) = \log(3)
\][/tex]
[tex]\[
\frac{1}{4} \log 81 = \log(81^{1/4}) = \log(\sqrt[4]{81}) = \log(3)
\][/tex]
[tex]\[
2 \log 6 = \log(6^2) = \log(36)
\][/tex]
Now we have:
[tex]\[
\log(3) + \log(3) + \log(36) - \log(12)
\][/tex]
Combine the logarithms:
[tex]\[
\log(3) + \log(3) = \log(3 \times 3) = \log(9)
\][/tex]
[tex]\[
\log(9) + \log(36) = \log(9 \times 36) = \log(324)
\][/tex]
[tex]\[
\log(324) - \log(12) = \log\left(\frac{324}{12}\right) = \log(27)
\][/tex]
Thus, the result is:
[tex]\[
\log(27)
\][/tex]
