Two unrelated log parts: one non-log algebraic part

1. (i) Using the laws of logarithms, solve

$$\log _ { 3 } ( 4 x ) + 2 = \log _ { 3 } ( 5 x + 7 )$$ (ii) Given that $$\sum _ { r = 1 } ^ { 2 } \log _ { a } \left( y ^ { r } \right) = \sum _ { r = 1 } ^ { 2 } \left( \log _ { a } y \right) ^ { r } \quad y > 1 , a > 1 , y \neq a$$ find \(y\) in terms of \(a\), giving your answer in simplest form.

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1. Find the value of \(x\) in each of the following:
   1. \(\quad \log _ { 9 } x = 0\);
   2. \(\quad \log _ { 9 } x = \frac { 1 } { 2 }\).
2. Given that $$2 \log _ { a } n = \log _ { a } 18 + \log _ { a } ( n - 4 )$$ find the possible values of \(n\).

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1. Given that \(2 \log _ { k } x - \log _ { k } 5 = 1\), express \(k\) in terms of \(x\). Give your answer in a form not involving logarithms.
2. Given that \(\log _ { a } y = \frac { 3 } { 2 }\) and that \(\log _ { 4 } a = b + 2\), show that \(y = 2 ^ { p }\), where \(p\) is an expression in terms of \(b\).
   \includegraphics[max width=\textwidth, alt={}]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-09_2102_1717_605_150}

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1. Sketch the graph of \(y = \frac { 1 } { 2 ^ { x } }\), indicating the value of the intercept on the \(y\)-axis.
2. Use logarithms to solve the equation \(\frac { 1 } { 2 ^ { x } } = \frac { 5 } { 4 }\), giving your answer to three significant figures.
3. Given that $$\log _ { a } \left( b ^ { 2 } \right) + 3 \log _ { a } y = 3 + 2 \log _ { a } \left( \frac { y } { a } \right)$$ express \(y\) in terms of \(a\) and \(b\).
   Give your answer in a form not involving logarithms.

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1. Solve the equation \(3 \log _ { a } x = \log _ { a } 8\).
2. Show that $$3 \log _ { a } 6 - \log _ { a } 8 = \log _ { a } 27$$
3. 1. The point \(P ( 3 , p )\) lies on the curve \(y = 3 \log _ { 10 } x - \log _ { 10 } 8\). Show that \(p = \log _ { 10 } \left( \frac { 27 } { 8 } \right)\).
   2. The point \(Q ( 6 , q )\) also lies on the curve \(y = 3 \log _ { 10 } x - \log _ { 10 } 8\). Show that the gradient of the line \(P Q\) is \(\log _ { 10 } 2\).

Given that \(2^x = \frac{1}{\sqrt{2}}\) and \(2^y = 4\sqrt{2}\),

1. find the exact value of \(x\) and the exact value of \(y\), [3]
2. calculate the exact value of \(2^{y-x}\). [2]

The sequence \(u_1, u_2, u_3, \ldots, u_n\) is defined by the recurrence relation $$u_{n+1} = pu_n + 5, \quad u_1 = 2, \text{ where } p \text{ is a constant.}$$ Given that \(u_3 = 8\),

1. show that one possible value of \(p\) is \(\frac{1}{2}\) and find the other value of \(p\). [5]

Using \(p = \frac{1}{2}\),

1. write down the value of \(\log_2 p\). [1]

Given also that \(\log_2 q = t\),

1. express \(\log_2 \left(\frac{p^3}{\sqrt{q}}\right)\) in terms of \(t\). [3]

The sequence \(u_1, u_2, u_3, \ldots, u_n\) is defined by the recurrence relation $$u_{n+1} = pu_n + 5, u_1 = 2, \text{ where } p \text{ is a constant.}$$ Given that \(u_3 = 8\),

1. show that one possible value of \(p\) is \(\frac{1}{2}\) and find the other value of \(p\). [5]

Using \(p = \frac{1}{2}\),

1. write down the value of \(\log_2 p\). [1]

Given also that \(\log_2 q = t\),

1. express \(\log_2 \left(\frac{p^3}{\sqrt{q}}\right)\) in terms of \(t\). [3]

It is given that $$\ln x - \ln y = 3$$

1. Express \(x\) in terms of \(y\) in a form not involving logarithms. [3 marks]
2. Given also that $$x + y = 10$$ find the exact value of \(y\) and the exact value of \(x\) [3 marks]

Solve the following equations for values of \(x\).

1. \(\ln(2x + 5) = 3\) [2]
2. \(5^{2x+1} = 14\) [3]
3. \(3\log_7(2x) - \log_7(8x^2) + \log_7 x = \log_3 81\) [6]

\includegraphics{figure_6} **In this question you must show all stages of your working.** **Solutions relying on calculator technology are not acceptable.** Figure 6 shows a sketch of part of the curve with equation $$y = 3 \times 2^{2x}$$ The point \(P\left(a, 96\sqrt{2}\right)\) lies on the curve.

1. Find the exact value of \(a\). [3]

The curve with equation \(y = 3 \times 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).

1. Show that the \(x\) coordinate of \(Q\) is $$\frac{3 + 2\log_2 3}{3 + \log_2 3}$$ [5]

\includegraphics{figure_6} In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 6 shows a sketch of part of the curve with equation $$y = 3x \cdot 2^{2x}.$$ The point \(P(a, 96\sqrt{2})\) lies on the curve.

1. Find the exact value of \(a\). [3]

The curve with equation \(y = 3x \cdot 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).

1. Show that the \(x\) coordinate of \(Q\) is \(\frac{3 + 2\log_2 3}{3 + \log_2 3}\). [4]