Laws of Logarithms

2. Find the values of \(x\) such that $$2 \log _ { 3 } x - \log _ { 3 } ( x - 2 ) = 2$$

3

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 \(n\).

1．（a）Solve the equation $$\sqrt { } ( 3 x + 16 ) = 3 + \sqrt { } ( x + 1 )$$ （b）Solve the equation $$\log _ { 3 } ( x - 7 ) - \frac { 1 } { 2 } \log _ { 3 } x = 1 - \log _ { 3 } 2$$

1 Given that \(2 \ln ( x + 4 ) - \ln x = \ln ( x + a )\), express \(x\) in terms of \(a\).

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1. Given that $$\log _ { a } x = \log _ { a } 16 - \log _ { a } 2$$ write down the value of \(x\).
2. Given that $$\log _ { a } y = 2 \log _ { a } 3 + \log _ { a } 4 + 1$$ express \(y\) in terms of \(a\), giving your answer in a form not involving logarithms.

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.

9. Given that \(y = 3 x ^ { 2 }\),

1. show that \(\log _ { 3 } y = 1 + 2 \log _ { 3 } x\)
2. Hence, or otherwise, solve the equation $$1 + 2 \log _ { 3 } x = \log _ { 3 } ( 28 x - 9 )$$

2

1. Show that the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$ can be written as a quadratic equation in \(x\).
2. Hence solve the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$

6. Given that $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$

1. show that $$4 x ^ { 2 } - 16 x - 9 = 0$$
2. Hence solve the equation $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$

2 Solve the equation \(\ln ( 3 - 2 x ) - 2 \ln x = \ln 5\).

1 Solve the equation $$\ln ( 3 x + 4 ) = 2 \ln ( x + 1 )$$ giving your answer correct to 3 significant figures.

2 Solve the equation \(\ln ( 3 - 2 x ) - 2 \ln x = \ln 5\).

18. Given that \(p = \log _ { q } 16\), express in terms of \(p\),

1. \(\log _ { q } 2\),
2. \(\log _ { q } ( 8 q )\).

13. Given that \(p = \log _ { a } 9\) and \(q = \log _ { a } 10\), where \(a\) is a constant, find in terms of \(p\) and \(q\),

1. \(\log _ { a } 900\)
2. \(\log _ { a } 0.3\)

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11.

1. Given that \(\log _ { 3 } c = m\) and \(\log _ { 27 } d = n\), express \(\frac { \sqrt { c } } { d ^ { 2 } }\) in the form \(3 ^ { y }\), where \(y\) is an expression in terms of \(m\) and \(n\).
2. Show that the equation $$\log _ { 4 } ( 2 x + 3 ) + \log _ { 4 } ( 2 x + 15 ) = 1 + \log _ { 4 } ( 14 x + 5 )$$ has only one solution and state its value.

1 Solve the equation $$\ln ( x + 1 ) - \ln x = 2 \ln 2$$

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1. 1. Express \(\log _ { 3 } ( 4 x + 7 ) - \log _ { 3 } x\) as a single logarithm.
   2. Hence solve the equation \(\log _ { 3 } ( 4 x + 7 ) - \log _ { 3 } x = 2\).
2. Use the trapezium rule, with two strips of width 3, to find an approximate value for $$\int _ { 3 } ^ { 9 } \log _ { 10 } x \mathrm {~d} x ,$$ giving your answer correct to 3 significant figures.

2 Solve the equation \(\ln ( x - 5 ) = 7 - \ln x\). Give your answer correct to 2 decimal places.

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

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

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.
Do not use staples, paper clips, glue or correction fluid.
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.

7

1. 1. Find the equation of the straight line in the form $$\log _ { 10 } T = a + b \log _ { 10 } d$$ where \(a\) and \(b\) are constants to be found.
      7
2. (ii) Show that $$T = \mathrm { K } d ^ { \mathrm { n } }$$ where K and n are constants to be found.
   7
3. Neptune takes approximately 60000 days to complete one orbit of the Sun.
   Use your answer to 7(a)(ii) to find an estimate for the average distance of Neptune from the Sun.

7 Simplify

1. \(\log _ { 10 } x ^ { 5 } + 3 \log _ { 10 } x ^ { 4 }\),
2. \(\log _ { a } 1 - \log _ { a } a ^ { b }\).

3 Express as a single logarithm $$2 \log _ { a } 6 - \log _ { a } 3$$

2 \log _ { 2 } y = 5 - \log _ { 2 } x
\log _ { x } y = - 3 \end{gathered}$$ for \(x > 0 , y > 0\)

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

5. (a) Given that \(t = \log _ { 3 } x\), find expressions in terms of \(t\) for

1. \(\log _ { 3 } x ^ { 2 }\),
2. \(\log _ { 9 } x\).
   (b) Hence, or otherwise, find to 3 significant figures the value of \(x\) such that $$\log _ { 3 } x ^ { 2 } - \log _ { 9 } x = 4 .$$

5. A student is asked to solve the equation $$\log _ { 3 } x - \log _ { 3 } \sqrt { x - 2 } = 1$$ The student's attempt is shown $$\begin{aligned} \log _ { 3 } x - \log _ { 3 } \sqrt { x - 2 } & = 1 \\ x - \sqrt { x - 2 } & = 3 ^ { 1 } \\ x - 3 & = \sqrt { x - 2 } \\ ( x - 3 ) ^ { 2 } & = x - 2 \\ x ^ { 2 } - 7 x + 11 & = 0 \\ x = \frac { 7 + \sqrt { 5 } } { 2 } \quad \text { or } \quad x & = \frac { 7 - \sqrt { 5 } } { 2 } \end{aligned}$$ a. Identify the error made by this student, giving a brief explanation.
b. Write out the correct solution.

12. Prove that $$\log _ { 7 } a \times \log _ { a } 19 = \log _ { 7 } 19$$ whatever the value of the positive constant \(a\).

2. \begin{figure}[h] \end{figure} Figure 1 shows the linear relationship between \(\log _ { 6 } T\) and \(\log _ { 6 } x\) The line passes through the points \(( 0,4 )\) and \(( 2,0 )\) as shown.

1. 1. Find an equation linking \(\log _ { 6 } T\) and \(\log _ { 6 } x\)
   2. Hence find the exact value of \(T\) when \(x = 216\)
2. Find an equation, not involving logs, linking \(T\) with \(x\)

7 Express \(\log _ { a } x ^ { 3 } + \log _ { a } \sqrt { x }\) in the form \(k \log _ { a } x\).

6. Given that the equation $$2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3 ,$$ only has one solution, find the value of \(x\).
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1

1. Solve the equation \(\ln ( 2 + x ) - \ln x = 2 \ln 3\).
2. Hence solve the equation \(\ln ( 2 + \cot y ) - \ln ( \cot y ) = 2 \ln 3\) for \(0 < y < \frac { 1 } { 2 } \pi\). Give your answer correct to 4 significant figures.

6. Find the values of \(x\) such that $$2 \log _ { 3 } x - \log _ { 3 } ( x - 2 ) = 2$$ [BLANK PAGE]

9. a) Write the following as a single logarithm $$3 \log ( x ) - \frac { 1 } { 2 } \log ( y ) + 2$$ b) Solve \(2 ^ { x } e ^ { 3 x + 1 } = 10\) Giving your answer to (b) in the form \(\frac { \ln a + b } { \ln c + d }\), where \(a , b , c\) and \(d\) are integers.
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