1.06f Laws of logarithms: addition, subtraction, power rules

453 questions

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OCR C2 2006 January Q7
8 marks Moderate -0.5
7
  1. Express each of the following in terms of \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
    1. \(\log _ { 10 } \left( \frac { x } { y } \right)\)
    2. \(\log _ { 10 } \left( 10 x ^ { 2 } y \right)\)
    3. Given that $$2 \log _ { 10 } \left( \frac { x } { y } \right) = 1 + \log _ { 10 } \left( 10 x ^ { 2 } y \right)$$ find the value of \(y\) correct to 3 decimal places.
OCR C2 2008 January Q3
4 marks Easy -1.8
3 Express each of the following as a single logarithm:
  1. \(\log _ { a } 2 + \log _ { a } 3\),
  2. \(2 \log _ { 10 } x - 3 \log _ { 10 } y\).
OCR C2 2005 June Q7
7 marks Moderate -0.8
7
  1. Evaluate \(\log _ { 5 } 15 + \log _ { 5 } 20 - \log _ { 5 } 12\).
  2. Given that \(y = 3 \times 10 ^ { 2 x }\), show that \(x = a \log _ { 10 } ( b y )\), where the values of the constants \(a\) and \(b\) are to be found.
OCR C2 2006 June Q9
11 marks Moderate -0.8
9
  1. Sketch the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), and state the coordinates of any point where the curve crosses an axis.
  2. Use the trapezium rule, with 4 strips of width 0.5 , to estimate the area of the region bounded by the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), the axes, and the line \(x = 2\).
  3. The point \(P\) on the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) has \(y\)-coordinate equal to \(\frac { 1 } { 6 }\). Prove that the \(x\)-coordinate of \(P\) may be written as $$1 + \frac { \log _ { 10 } 3 } { \log _ { 10 } 2 }$$
OCR C2 2007 June Q3
5 marks Moderate -0.8
3 U se logarithms to solve the equation \(3 ^ { 2 x + 1 } = 5 ^ { 200 }\), giving the value of \(x\) correct to 3 significant figures.
OCR C2 2007 June Q9
14 marks Standard +0.3
9 The polynomial \(f ( x )\) is given by $$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4 .$$
  1. (a) Show that ( \(\mathrm { x } + 1\) ) is a factor of \(\mathrm { f } ( \mathrm { x } )\).
    (b) Hence find the exact roots of the equation \(f ( x ) = 0\).
  2. (a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form \(f ( x ) = 0\).
    (b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root.
OCR MEI C2 2005 January Q8
5 marks Moderate -0.8
8
  1. Solve the equation \(10 ^ { x } = 316\).
  2. Simplify \(\log _ { a } \left( a ^ { 2 } \right) - 4 \log _ { a } \left( \frac { 1 } { a } \right)\).
OCR MEI C2 2007 January Q10
4 marks Easy -1.2
10
  1. Express \(\log _ { a } x ^ { 4 } + \log _ { a } \left( \frac { 1 } { x } \right)\) as a multiple of \(\log _ { a } x\).
  2. Given that \(\log _ { 10 } b + \log _ { 10 } c = 3\), find \(b\) in terms of \(c\).
OCR MEI C2 2008 January Q9
4 marks Moderate -0.3
9 You are given that \(\log _ { 10 } y = 3 x + 2\).
  1. Find the value of \(x\) when \(y = 500\), giving your answer correct to 2 decimal places.
  2. Find the value of \(y\) when \(x = - 1\).
  3. Express \(\log _ { 10 } \left( y ^ { 4 } \right)\) in terms of \(x\).
  4. Find an expression for \(y\) in terms of \(x\). Section B (36 marks)
OCR MEI C2 2005 June Q5
5 marks Easy -1.8
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 \(\log _ { a } x\).
OCR MEI C2 2005 June Q11
10 marks Standard +0.3
11 There is a flowerhead at the end of each stem of an oleander plant. The next year, each flowerhead is replaced by three stems and flowerheads, as shown in Fig. 11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{faeaf2aa-ed4e-4926-b402-40c4c9aad479-5_501_1102_431_504} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. How many flowerheads are there in year 5?
  2. How many flowerheads are there in year \(n\) ?
  3. As shown in Fig. 11, the total number of stems in year 2 is 4, (that is, 1 old one and 3 new ones). Similarly, the total number of stems in year 3 is 13 , (that is, \(1 + 3 + 9\) ). Show that the total number of stems in year \(n\) is given by \(\frac { 3 ^ { n } - 1 } { 2 }\).
  4. Kitty's oleander has a total of 364 stems. Find
    (A) its age,
    (B) how many flowerheads it has.
  5. Abdul's oleander has over 900 flowerheads. Show that its age, \(y\) years, satisfies the inequality \(y > \frac { \log _ { 10 } 900 } { \log _ { 10 } 3 } + 1\).
    Find the smallest integer value of \(y\) for which this is true.
OCR MEI C2 2007 June Q6
5 marks Moderate -0.8
6
  1. Write down the values of \(\log _ { a } 1\) and \(\log _ { a } a\), where \(a > 1\).
  2. Show that \(\log _ { a } x ^ { 10 } - 2 \log _ { a } \left( \frac { x ^ { 3 } } { 4 } \right) = 4 \log _ { a } ( 2 x )\).
OCR MEI C2 2007 June Q7
5 marks Easy -1.2
7
  1. Sketch the graph of \(y = 3 ^ { x }\).
  2. Use logarithms to solve the equation \(3 ^ { x } = 20\). Give your answer correct to 2 decimal places.
OCR MEI C2 2007 June Q11
12 marks Moderate -0.3
11
  1. André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
    1. How many counters are there in his sixth pile?
    2. André makes ten piles of counters. How many counters has he used altogether?
  2. In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start. The probability \(\mathrm { P } _ { n }\) of Betty starting on her \(n\)th throw is given by $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
    1. Calculate \(\mathrm { P } _ { 4 }\). Give your answer as a fraction.
    2. The values \(\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots\) form an infinite geometric progression. State the first term and the common ratio of this progression. Hence show that \(\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1\).
    3. Given that \(\mathrm { P } _ { n } < 0.001\), show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$ Hence find the least value of \(n\) for which \(\mathrm { P } _ { n } < 0.001\).
OCR MEI C2 2009 June Q9
3 marks Easy -1.2
9 Simplify
  1. \(10 - 3 \log _ { a } a\),
  2. \(\frac { \log _ { 10 } a ^ { 5 } + \log _ { 10 } \sqrt { a } } { \log _ { 10 } a }\). Section B (36 marks)
OCR MEI C2 Q2
5 marks Easy -1.2
2
  1. Write \(\log _ { 2 } 5 + \log _ { 2 } 1.6\) as an integer.
  2. Solve the equation \(2 ^ { x } = 3\), giving your answer correct to 4 decimal places.
OCR MEI C2 Q3
4 marks Moderate -0.8
3
  1. Write \(\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 )\) as a single logarithm.
  2. Without using your calculator, verify that \(x = 4\) is a root of the equation $$\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 ) = 1$$
OCR C2 Q7
10 marks Standard +0.3
  1. Given that $$\log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x$$ show that $$y = 2 x + 1$$
  2. Solve the simultaneous equations $$\begin{aligned} & \log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x \\ & 2 \log _ { 3 } y = 2 + \log _ { 3 } x \end{aligned}$$
OCR C2 Q5
8 marks Moderate -0.3
5. (a) Given that \(t = \log _ { 3 } x\),
  1. write down an expression in terms of \(t\) for \(\log _ { 3 } x ^ { 2 }\),
  2. show that \(\log _ { 9 } x = \frac { 1 } { 2 } t\).
    (b) Hence, or otherwise, find to 3 significant figures the value of \(x\) such that $$\log _ { 3 } x ^ { 2 } - \log _ { 9 } x = 4$$
OCR C2 Q5
9 marks Moderate -0.3
5.
  1. Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$
  2. Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0$$
OCR C2 Q1
4 marks Moderate -0.3
  1. Solve the equation
$$\log _ { 5 } ( 4 x + 3 ) - \log _ { 5 } ( x - 1 ) = 2$$
OCR C2 Q7
10 marks Standard +0.3
7. The second and third terms of a geometric series are \(\log _ { 3 } 4\) and \(\log _ { 3 } 16\) respectively.
  1. Find the common ratio of the series.
  2. Show that the first term of the series is \(\log _ { 3 } 2\).
  3. Find, to 3 significant figures, the sum of the first six terms of the series.
OCR C2 Q5
7 marks Moderate -0.3
5.
  1. Find the value of \(a\) such that $$\log _ { a } 27 = 3 + \log _ { a } 8$$
  2. Solve the equation $$2 ^ { x + 3 } = 6 ^ { x - 1 }$$ giving your answer to 3 significant figures.
OCR C3 Q2
7 marks Moderate -0.3
2. Solve each equation, giving your answers in exact form.
  1. \(\quad \ln ( 2 x - 3 ) = 1\)
  2. \(3 \mathrm { e } ^ { y } + 5 \mathrm { e } ^ { - y } = 16\)
OCR C3 Q5
7 marks Moderate -0.3
5. The function \(f\) is defined by $$\mathrm { f } ( x ) \equiv 4 - \ln 3 x , \quad x \in \mathbb { R } , \quad x > 0$$
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Sketch the curve \(y = \mathrm { f } ( x )\). The function g is defined by $$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
  3. Show that $$\operatorname { fg } ( x ) = x + a - \ln b$$ where \(a\) and \(b\) are integers to be found.