State value of basic log

1 Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9) \section*{READ THESE INSTRUCTIONS FIRST} Write your Centre number, candidate number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
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Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50. This document consists of 11 printed pages and 1 blank page. 1 Solve the equation \(\ln ( 3 x + 1 ) - \ln ( x + 2 ) = 1\), giving your answer in terms of e.

5

1. Write down the value of \(\log _ { 5 } 5\).
2. Find \(\log _ { 3 } \left( \frac { 1 } { 9 } \right)\).
3. Express \(\log _ { a } x + \log _ { a } \left( x ^ { 5 } \right)\) as a multiple of \(\log _ { a } x\).

1 Write down the values of \(\log _ { a } a\) and \(\log _ { a } \left( a ^ { 3 } \right)\).

9

1. State the value of \(\log _ { a } a\).
2. Express each of the following in terms of \(\log _ { a } x\).
   (A) \(\log _ { a } x ^ { 3 } + \log _ { a } \sqrt { x }\) (B) \(\log _ { a } \frac { 1 } { x }\) Section B (36 marks)

4 Given that \(a > 0\), state the values of

1. \(\log _ { a } 1\),
2. \(\log _ { a } \left( a ^ { 3 } \right) ^ { 6 }\),
3. \(\log _ { a } \sqrt { a }\).

1 Write down the value of
(A) \(\log _ { a } \left( a ^ { 4 } \right)\),
(B) \(\log _ { a } \left( \frac { 1 } { a } \right)\).

1 The diagram shows the curve with equation \(y = \ln ( 6 x )\). \includegraphics[max width=\textwidth, alt={}, center]{7148f43d-dc7d-43e2-b96e-ed1fb94073bf-2_448_501_370_790}

1. State the \(x\)-coordinate of the point of intersection of the curve with the \(x\)-axis.
   (1 mark)
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (2 marks)
3. Use Simpson's rule with 6 strips (7 ordinates) to find an estimate for \(\int _ { 1 } ^ { 7 } \ln ( 6 x ) \mathrm { d } x\), giving your answer to three significant figures.

1 Simplify \(\log _ { 2 } 8 ^ { a }\) Circle your answer. \(a ^ { 3 }\) \(2 a\) 3a \(8 a\)

3 Find the value of \(\log _ { a } \left( a ^ { 3 } \right) + \log _ { a } \left( \frac { 1 } { a } \right)\) [0pt] [2 marks]