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

3. Given that \(p = \log _ { 2 } 3\) and \(q = \log _ { 2 } 5\), find expressions in terms of \(p\) and \(q\) for

1. \(\quad \log _ { 2 } 45\),
2. \(\quad \log _ { 2 } 0.3\)

4. Solve each equation, giving your answers to an appropriate degree of accuracy.

1. \(3 ^ { x - 2 } = 5\)
2. \(\quad \log _ { 2 } ( 6 - y ) = 3 - \log _ { 2 } y\)

8.

1. Given that $$\log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x ,$$ show that $$y = 2 x + 1 .$$
2. Solve the simultaneous equations $$\begin{aligned} & \log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x \\ & 2 \log _ { 3 } y = 2 + \log _ { 3 } x \end{aligned}$$

8. 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.

3. (a) Given that \(y = \log _ { 2 } x\), find expressions in terms of \(y\) for

1. \(\quad \log _ { 2 } \left( \frac { x } { 2 } \right)\),
2. \(\log _ { 2 } ( \sqrt { x } )\).
   (b) Hence, or otherwise, solve the equation $$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$

5

1. Solve the equation \(2 \mathrm { e } ^ { x } = 5\), giving your answer as an exact natural logarithm.
2. 1. By substituting \(y = \mathrm { e } ^ { x }\), show that the equation \(2 \mathrm { e } ^ { x } + 5 \mathrm { e } ^ { - x } = 7\) can be written as $$2 y ^ { 2 } - 7 y + 5 = 0$$
   2. Hence solve the equation \(2 \mathrm { e } ^ { x } + 5 \mathrm { e } ^ { - x } = 7\), giving your answers as exact values of \(x\).

5

1. Given that \(2 \ln x = 5\), find the exact value of \(x\).
2. Solve the equation $$2 \ln x + \frac { 15 } { \ln x } = 11$$ giving your answers as exact values of \(x\).

3. Solve each equation, giving your answers in exact form.

1. \(\quad \ln ( 2 x - 3 ) = 1\)
2. \(3 \mathrm { e } ^ { y } + 5 \mathrm { e } ^ { - y } = 16\)

3. (a) Given that \(y = \ln x\), find expressions in terms of \(y\) for

1. \(\quad \log _ { 2 } x\),
2. \(\ln \frac { x ^ { 2 } } { \mathrm { e } }\).
   (b) Hence, or otherwise, solve the equation $$\log _ { 2 } x = 4 - \ln \frac { x ^ { 2 } } { \mathrm { e } } ,$$ giving your answer to 2 decimal places.

3.

1. Simplify $$\frac { 2 x ^ { 2 } + 3 x - 9 } { 2 x ^ { 2 } - 7 x + 6 }$$
2. Solve the equation $$\ln \left( 2 x ^ { 2 } + 3 x - 9 \right) = 2 + \ln \left( 2 x ^ { 2 } - 7 x + 6 \right)$$ giving your answer in terms of e.
   

5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv 6 x - 1 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} ( x ) \equiv \log _ { 2 } ( 3 x + 1 ) , \quad x \in \mathbb { R } , \quad x > - \frac { 1 } { 3 } \end{aligned}$$

1. Evaluate \(\operatorname { gf } ( 1 )\).
2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
3. Find, in terms of natural logarithms, the solution of the equation $$\mathrm { fg } ^ { - 1 } ( x ) = 2$$

4 A painting was valued on 1 April 2001 at \(\pounds 5000\).
The value of this painting is modelled by $$V = A p ^ { t }$$ where \(\pounds V\) is the value \(t\) years after 1 April 2001, and \(A\) and \(p\) are constants.

1. Write down the value of \(A\).
2. According to the model, the value of this painting on 1 April 2011 was \(\pounds 25000\). Using this model:
   1. show that \(p ^ { 10 } = 5\);
   2. use logarithms to find the year in which the painting will be valued at \(\pounds 75000\).
3. A painting by another artist was valued at \(\pounds 2500\) on 1 April 1991. The value of this painting is modelled by $$W = 2500 q ^ { t }$$ where \(\pounds W\) is the value \(t\) years after 1 April 1991, and \(q\) is a constant.
   1. Show that, according to the two models, the value of the two paintings will be the same \(T\) years after 1 April 1991, $$\text { where } T = \frac { \ln \left( \frac { 5 } { 2 } \right) } { \ln \left( \frac { p } { q } \right) }$$
   2. Given that \(p = 1.029 q\), find the year in which the two paintings will have the same value.
      [0pt] [1 mark]

7

1. The equation \(2 x ^ { 3 } + 5 x ^ { 2 } + 3 x - 132000 = 0\) has exactly one real root \(\alpha\).
   1. Show that \(\alpha\) lies in the interval \(39 < \alpha < 40\).
   2. Taking \(x _ { 1 } = 40\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\). Give your answer to two decimal places.
2. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$\sum _ { r = 1 } ^ { n } 2 r ( 3 r + 2 ) = n ( n + p ) ( 2 n + q )$$ where \(p\) and \(q\) are integers.
3. 1. Express \(\log _ { 8 } 4 ^ { r }\) in the form \(\lambda r\), where \(\lambda\) is a rational number.
   2. By first finding a suitable cubic inequality for \(k\), find the greatest value of \(k\) for which \(\sum _ { r = k + 1 } ^ { 60 } ( 3 r + 2 ) \log _ { 8 } 4 ^ { r }\) is greater than 106060.
      [0pt] [4 marks]

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.}

1. Use the substitution \(t = \tan \left( \frac { \theta } { 2 } \right)\) to show that $$\int \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \int \frac { a } { ( t + b ) ^ { 2 } + c } d t$$ where \(a\), \(b\) and \(c\) are constants to be determined.
2. Hence show that $$\int _ { \frac { \pi } { 2 } } ^ { \frac { 2 \pi } { 3 } } \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \ln \left( \frac { 2 \sqrt { 3 } } { 3 } \right)$$

5.[In this question the values of \(a , x\) ,and \(n\) are such that \(a\) and \(x\) are positive real numbers,with \(a > 1 , x \neq a , x \neq 1\) and \(n\) is an integer with \(n > 1\) ] Sam was confused about the rules of logarithms and thought that $$\log _ { a } x ^ { n } = \left( \log _ { a } x \right) ^ { n }$$

1. Given that \(x\) satisfies statement(1)find \(x\) in terms of \(a\) and \(n\) . Sam also thought that $$\log _ { a } x + \log _ { a } x ^ { 2 } + \ldots + \log _ { a } x ^ { n } = \log _ { a } x + \left( \log _ { a } x \right) ^ { 2 } + \ldots + \left( \log _ { a } x \right) ^ { n }$$
2. For \(n = 3 , x _ { 1 }\) and \(x _ { 2 } \left( x _ { 1 } > x _ { 2 } \right)\) are the two values of \(x\) that satisfy statement(2).
   1. Find,in terms of \(a\) ,an expression for \(x _ { 1 }\) and an expression for \(x _ { 2 }\) .
   2. Find the exact value of \(\log _ { a } \left( \frac { x _ { 1 } } { x _ { 2 } } \right)\) .
3. Show that if \(\log _ { a } x\) satisfies statement(2)then $$2 \left( \log _ { a } x \right) ^ { n } - n ( n + 1 ) \log _ { a } x + \left( n ^ { 2 } + n - 2 \right) = 0$$

4 In this question you must show detailed reasoning. The cubic polynomial \(6 x ^ { 3 } + k x ^ { 2 } + 57 x - 20\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).

1. Use the factor theorem to show that \(k = - 37\).
2. Using this value of \(k\), factorise \(\mathrm { f } ( x )\) completely.
3. 1. Hence find the three values of \(t\) satisfying the equation \(6 \mathrm { e } ^ { - 3 t } - 37 \mathrm { e } ^ { - 2 t } + 57 \mathrm { e } ^ { - t } - 20 = 0\).
   2. Express the sum of the three values found in part (c)(i) as a single logarithm.

5 A student was asked to solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\). The student's attempt is written out below. $$\begin{aligned} & 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0 \\ & 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x = 2 \\ & x = 8 \end{aligned}$$

1. Identify the two mistakes that the student has made.
2. Solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\), giving your answers in an exact form.

9

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 \(\log _ { 10 } 2\).

6 The diagram shows a sketch of the curve with equation \(y = 3 \left( 2 ^ { x } + 1 \right)\). \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-5_465_851_390_607} The curve \(y = 3 \left( 2 ^ { x } + 1 \right)\) intersects the \(y\)-axis at the point \(A\).

1. Find the \(y\)-coordinate of the point \(A\).
2. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 6 } 3 \left( 2 ^ { x } + 1 \right) d x\).
3. The line \(y = 21\) intersects the curve \(y = 3 \left( 2 ^ { x } + 1 \right)\) at the point \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$2 ^ { x } = 6$$
   2. Use logarithms to find the \(x\)-coordinate of \(P\), giving your answer to three significant figures.

8

1. It is given that \(n\) satisfies the equation $$\log _ { a } n = \log _ { a } 3 + \log _ { a } ( 2 n - 1 )$$ Find the value of \(n\).
2. Given that \(\log _ { a } x = 3\) and \(\log _ { a } y - 3 \log _ { a } 2 = 4\) :
   1. express \(x\) in terms of \(a\);
   2. express \(x y\) in terms of \(a\).

2. Given that \(p = \log _ { 2 } 3\) and \(q = \log _ { 2 } 5\), find expressions in terms of \(p\) and \(q\) for

1. \(\quad \log _ { 2 } 45\),
2. \(\log _ { 2 } 0.3\)

4 Show that, for \(x > 0\) $$\log _ { 10 } \frac { x ^ { 4 } } { 100 } + \log _ { 10 } 9 x - \log _ { 10 } x ^ { 3 } \equiv 2 \left( - 1 + \log _ { 10 } 3 x \right)$$

6 Show that the equation $$\begin{aligned} & \qquad 2 \log _ { 10 } x = \log _ { 10 } 4 + \log _ { 10 } ( x + 8 ) \\ & \text { has exactly one solution. } \\ & \text { Fully justify your answer. } \end{aligned}$$