Simplify or prove logarithmic identity

1 Show that \(\ln \left( x ^ { 3 } - 4 x \right) - \ln \left( x ^ { 2 } - 2 x \right) \equiv \ln ( x + 2 )\).

6

1. Write down the values of \(\log _ { a } 1\) and \(\log _ { a } a\), where \(a > 1\).
2. Show that \(\log _ { a } x ^ { 10 } - 2 \log _ { a } \left( \frac { x ^ { 3 } } { 4 } \right) = 4 \log _ { a } ( 2 x )\).

6．（a）Show that $$\sqrt { 2 + \sqrt { 3 } } - \sqrt { 2 - \sqrt { 3 } } = \sqrt { 2 }$$ （b）Hence prove that $$\log _ { \frac { 1 } { 8 } } ( \sqrt { 2 + \sqrt { 3 } } - \sqrt { 2 - \sqrt { 3 } } ) = - \frac { 1 } { 6 } .$$ （c）Find all possible pairs of integers \(a\) and \(n\) such that $$\log _ { \frac { 1 } { n } } ( \sqrt { a + \sqrt { 15 } } - \sqrt { a - \sqrt { 15 } } ) = - \frac { 1 } { 2 } .$$

2 It is given that \(p = \mathrm { e } ^ { 280 }\) and \(q = \mathrm { e } ^ { 300 }\).

1. Use logarithm properties to show that \(\ln \left( \frac { \mathrm { e } \mathrm { p } ^ { 2 } } { q } \right) = 261\).
2. Find the smallest integer \(n\) which satisfies the inequality \(5 ^ { n } > p q\).

4 Show that, for \(x > 0\) $$\log _ { 10 } \frac { x ^ { 4 } } { 100 } + \log _ { 10 } 9 x - \log _ { 10 } x ^ { 3 } \equiv 2 \left( - 1 + \log _ { 10 } 3 x \right)$$

7 In this question you must show detailed reasoning.

1. Express \(\ln 3 \times \ln 9 \times \ln 27\) in terms of \(\ln 3\).
2. Hence show that \(\ln 3 \times \ln 9 \times \ln 27 > 6\).