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. Given that \(a\) is a positive constant and $$\int _ { a } ^ { 2 a } \frac { t + 1 } { t } \mathrm {~d} t = \ln 7$$ show that \(a = \ln k\), where \(k\) is a constant to be found.

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

9 Given that $$\log _ { 2 } x ^ { 3 } - \log _ { 2 } y ^ { 2 } = 9$$ show that $$x = A y ^ { p }$$ where \(A\) is an integer and \(p\) is a rational number. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-15_2488_1716_219_153}

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 \(\ln 3 \times \ln 9 \times \ln 27 > 6\).

It is given that \(y = 5^{x-1}\).

1. Show that \(x = 1 + \frac{\ln y}{\ln 5}\). [2]
2. Find an expression for \(\frac{dx}{dy}\) in terms of \(y\). [2]
3. Hence find the exact value of the gradient of the curve \(y = 5^{x-1}\) at the point \((3, 25)\). [2]

Given that \(a > 0\), determine which of these expressions is not equivalent to the others. Circle your answer. [1 mark] $$-2\log_{10}\left(\frac{1}{a}\right) \quad 2\log_{10}(a) \quad \log_{10}(a^2) \quad -4\log_{10}(\sqrt{a})$$

Prove that $$\log_a a \times \log_a 19 = \log_a 19$$ whatever the value of the positive constant \(a\). [3]

Show that $$\log_a(x^{10}) - 2\log_a\left(\frac{x^3}{4}\right) = 4\log_a(2x)$$ [3]

In the question you must show detailed reasoning Given that \(\log_a x = \frac{\log_n x}{\log_n a}\), show that the sum of the infinite series, where \(n = 0,1,2...\), $$\log_2 e - \log_4 e + \log_{16} e - \cdots + (-1)^n \log_{2^{2^n}} e + \cdots$$ is $$\frac{1}{\ln(2\sqrt{2})}$$ [5] [Total marks: 65]