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

8

1. Use logarithms to solve the equation $$2 ^ { n - 3 } = 18000$$ giving your answer correct to 3 significant figures.
2. Solve the simultaneous equations $$\log _ { 2 } x + \log _ { 2 } y = 8 , \quad \log _ { 2 } \left( \frac { x ^ { 2 } } { y } \right) = 7$$

4

1. Express \(2 \log _ { 3 } x - \log _ { 3 } ( x + 4 )\) as a single logarithm.
2. Hence solve the equation \(2 \log _ { 3 } x - \log _ { 3 } ( x + 4 ) = 2\).

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)

7 Simplify

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

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

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

7 Given that \(y = a + x ^ { b }\), find \(\log _ { 10 } x\) in terms of \(y\), \(a\) and \(b\).

8 Using logarithms, rearrange \(p = s t ^ { n }\) to make \(n\) the subject.

9 You are given that $$\log _ { a } x = \frac { 1 } { 2 } \log _ { a } 16 + \log _ { a } 75 - 2 \log _ { a } 5 .$$ Find the value of \(x\).

6 Fig. 6 shows the relationship between \(\log _ { 10 } x\) and \(\log _ { 10 } y\). \begin{figure}[h] \end{figure} Find \(y\) in terms of \(x\).

11 Jill has 3 daughters and no sons. They are generation 1 of Jill's descendants.
Each of her daughters has 3 daughters and no sons. Jill's 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3. Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.

1. How many of Jill's descendants would there be in generation 8 ?
2. How many of Jill's descendants would there be altogether in the first 15 generations?
3. After \(n\) generations, Jill would have over a million descendants altogether. Show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 2000003 } { \log _ { 10 } 3 } - 1 .$$ Hence find the least possible value of \(n\).
4. How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? \section*{END OF QUESTION PAPER}

2

1. Use Simpson's rule with four strips to find an approximation to $$\int _ { 4 } ^ { 12 } \ln x \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
2. Deduce an approximation to \(\int _ { 4 } ^ { 12 } \ln \left( x ^ { 10 } \right) \mathrm { d } x\).

1 Show that \(\int _ { \sqrt { 2 } } ^ { \sqrt { 6 } } \frac { 2 } { x } \mathrm {~d} x = \ln 3\).

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 \(5 ^ { n } > p q\).

6 The value of \(\int _ { 0 } ^ { 8 } \ln \left( 3 + x ^ { 2 } \right) \mathrm { d } x\) obtained by using Simpson's rule with four strips is denoted by \(A\).

1. Find the value of \(A\) correct to 3 significant figures.
2. Explain why an approximate value of \(\int _ { 0 } ^ { 8 } \ln \left( 9 + 6 x ^ { 2 } + x ^ { 4 } \right) \mathrm { d } x\) is \(2 A\).
3. Explain why an approximate value of \(\int _ { 0 } ^ { 8 } \ln \left( 3 \mathrm { e } + \mathrm { e } x ^ { 2 } \right) \mathrm { d } x\) is \(A + 8\).

8 The functions \(f\) and \(g\) are defined as follows: $$\begin{gathered} \mathrm { f } ( x ) = 2 + \ln ( x + 3 ) \text { for } x \geqslant 0 \\ \mathrm {~g} ( x ) = a x ^ { 2 } \text { for all real values of } x , \text { where } a \text { is a positive constant. } \end{gathered}$$

1. Given that \(\operatorname { gf } \left( \mathrm { e } ^ { 4 } - 3 \right) = 9\), find the value of \(a\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
3. Given that \(\mathrm { ff } \left( \mathrm { e } ^ { N } - 3 \right) = \ln \left( 53 \mathrm { e } ^ { 2 } \right)\), find the value of \(N\).

5 Given that \(y = \ln \left( \sqrt { \frac { 2 x - 1 } { 2 x + 1 } } \right)\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 x - 1 } - \frac { 1 } { 2 x + 1 }\).

2 Given that $$u _ { n } = \ln \left( \frac { 1 + x ^ { n + 1 } } { 1 + x ^ { n } } \right)$$ where \(x > - 1\), find \(\sum _ { n = 1 } ^ { N } u _ { n }\) in terms of \(N\) and \(x\). Find the sum to infinity of the series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ when

1. \(- 1 < x < 1\),
2. \(x = 1\).

9 For the sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), it is given that \(u _ { 1 } = 8\) and $$u _ { r + 1 } = \frac { 5 u _ { r } - 3 } { 4 }$$ for all \(r\).

1. Prove by mathematical induction that $$u _ { n } = 4 \left( \frac { 5 } { 4 } \right) ^ { n } + 3$$ for all positive integers \(n\).
2. Deduce the set of values of \(x\) for which the infinite series $$\left( u _ { 1 } - 3 \right) x + \left( u _ { 2 } - 3 \right) x ^ { 2 } + \ldots + \left( u _ { r } - 3 \right) x ^ { r } + \ldots$$ is convergent.
3. Use the result given in part (i) to find surds \(a\) and \(b\) such that $$\sum _ { n = 1 } ^ { N } \ln \left( u _ { n } - 3 \right) = N ^ { 2 } \ln a + N \ln b .$$

4 Evaluate the improper integral $$\int _ { 0 } ^ { \infty } \left( \frac { 2 x } { x ^ { 2 } + 4 } - \frac { 4 } { 2 x + 3 } \right) \mathrm { d } x$$ showing the limiting process used and giving your answer in the form \(\ln k\), where \(k\) is a constant.

7

1. It is given that \(y = \ln ( \cos x + \sin x )\).
   1. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 2 } { 1 + \sin 2 x }\).
   2. Find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
2. 1. Hence use Maclaurin's theorem to show that the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln ( \cos x + \sin x )\) are \(x - x ^ { 2 } + \frac { 2 } { 3 } x ^ { 3 }\).
   2. Write down the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln ( \cos x - \sin x )\).
3. Hence find the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \left( \frac { \cos 2 x } { \mathrm { e } ^ { 3 x - 1 } } \right)\).
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5

1. The graph of \(y = 2 ^ { x }\) can be transformed to the graph of \(y = 2 ^ { x + 4 }\) either by a translation or by a stretch.
   1. Give full details of the translation.
   2. Give full details of the stretch.
2. In this question you must show detailed reasoning. Solve the equation \(\log _ { 2 } ( 8 x ) = 1 - \log _ { 2 } ( 1 - x )\).

6. \begin{figure}[h] \end{figure} A company makes a particular type of children's toy.
The annual profit made by the company is modelled by the equation $$P = 100 - 6.25 ( x - 9 ) ^ { 2 }$$ where \(P\) is the profit measured in thousands of pounds and \(x\) is the selling price of the toy in pounds. A sketch of \(P\) against \(x\) is shown in Figure 1.
Using the model,

1. explain why \(\pounds 15\) is not a sensible selling price for the toy. Given that the company made an annual profit of more than \(\pounds 80000\)
2. find, according to the model, the least possible selling price for the toy. The company wishes to maximise its annual profit.
   State, according to the model,
3. 1. the maximum possible annual profit,
   2. the selling price of the toy that maximises the annual profit.

1. An advertising agency is monitoring the number of views of an online advert.

The equation $$\log _ { 10 } V = 0.072 t + 2.379 \quad 1 \leqslant t \leqslant 30 , t \in \mathbb { N }$$ is used to model the total number of views of the advert, \(V\), in the first \(t\) days after the advert went live.

1. Show that \(V = a b ^ { t }\) where \(a\) and \(b\) are constants to be found. Give the value of \(a\) to the nearest whole number and give the value of \(b\) to 3 significant figures.
2. Interpret, with reference to the model, the value of \(a b\). Using this model, calculate
3. the total number of views of the advert in the first 20 days after the advert went live. Give your answer to 2 significant figures.

1. The mass, \(A\) kg, of algae in a small pond, is modelled by the equation

$$A = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of weeks after the mass of algae was first recorded. Data recorded indicates that there is a linear relationship between \(t\) and \(\log _ { 10 } A\) given by the equation $$\log _ { 10 } A = 0.03 t + 0.5$$

1. Use this relationship to find a complete equation for the model in the form $$A = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 4 significant figures.
2. With reference to the model, interpret
   1. the value of the constant \(p\),
   2. the value of the constant \(q\).
3. Find, according to the model,
   1. the mass of algae in the pond when \(t = 8\), giving your answer to the nearest 0.5 kg ,
   2. the number of weeks it takes for the mass of algae in the pond to reach 4 kg .
4. State one reason why this may not be a realistic model in the long term.