Answer :
To simplify logarithmic expressions, we use the laws of logarithms, which include the product, quotient, and power rules.
Quotient Rule:
[tex]\log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N)[/tex]Power Rule:
[tex]\log_b(M^n) = n \log_b(M)[/tex]
Now, let's apply these rules to simplify each expression:
(i) [tex]\log_5 \left( \frac{11}{5} \right)[/tex]:
Apply the quotient rule:
[tex]\log_5 \left( \frac{11}{5} \right) = \log_5(11) - \log_5(5)[/tex]
(ii) [tex]\log_5 \left( \sqrt{8a^6} \right)[/tex]:
First, rewrite the square root as a power: [tex]\sqrt{8a^6} = (8a^6)^{1/2}[/tex].
Then apply the power rule:
[tex]\log_5 \left( \sqrt{8a^6} \right) = \log_5 \left( (8a^6)^{1/2} \right) = \frac{1}{2} \log_5(8a^6)[/tex]
Now, use the product rule inside the log:
[tex]\frac{1}{2} (\log_5(8) + \log_5(a^6))[/tex]
Then apply the power rule to [tex]a^6[/tex]:
[tex]\frac{1}{2} (\log_5(8) + 6\log_5(a))[/tex]
(iii) [tex]\ln \left( \frac{a^2b}{c} \right)[/tex]:
Apply the quotient rule:
[tex]\ln \left( \frac{a^2b}{c} \right) = \ln(a^2b) - \ln(c)[/tex]
Now apply the product rule:
[tex]\ln(a^2) + \ln(b) - \ln(c)[/tex]
Use the power rule on [tex]a^2[/tex]:
[tex]2\ln(a) + \ln(b) - \ln(c)[/tex]
(iv) [tex]\ln \left( \sqrt[3]{16x^3} \right)[/tex]:
First rewrite the cube root as a power: [tex]\sqrt[3]{16x^3} = (16x^3)^{1/3}[/tex].
Apply the power rule:
[tex]\ln \left( (16x^3)^{1/3} \right) = \frac{1}{3} \ln(16x^3)[/tex]
Apply the product rule inside the log:
[tex]\frac{1}{3} (\ln(16) + \ln(x^3))[/tex]
Use the power rule on [tex]x^3[/tex]:
[tex]\frac{1}{3} (\ln(16) + 3\ln(x))[/tex]
Simplify further:
[tex]\frac{1}{3} \ln(16) + \ln(x)[/tex]
(v) [tex]\log_2 \left( \frac{(1-a)^5}{b} \right)[/tex]:
Start with the quotient rule:
[tex]\log_2 \left( \frac{(1-a)^5}{b} \right) = \log_2((1-a)^5) - \log_2(b)[/tex]
Apply the power rule:
[tex]5\log_2(1-a) - \log_2(b)[/tex]