1.06c Logarithm definition: log_a(x) as inverse of a^x

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)

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

2 Fig. 9 shows the line \(y = x\) and the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } - 1 \right)\). The line and the curve intersect at the origin and at the point \(\mathrm { P } ( a , a )\). \begin{figure}[h] \end{figure}

1. Show that \(\mathrm { e } ^ { a } = 1 + 2 a\).
2. Show that the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = a\) is \(\frac { 1 } { 2 } a\). Hence find, in terms of \(a\), the area enclosed by the curve and the line \(y = x\).
3. Show that the inverse function of \(\mathrm { f } ( x )\) is \(\mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = \ln ( 1 + 2 x )\). Add a sketch of \(y = \mathrm { g } ( x )\) to the copy of Fig. 9.
4. Find the derivatives of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\). Hence verify that \(\mathrm { g } ^ { \prime } ( a ) = \frac { 1 } { \mathrm { f } ^ { \prime } ( a ) }\). Give a geometrical interpretation of this result.

1.(a)By writing \(u = \log _ { 4 } r\) ,where \(r > 0\) ,show that $$\log _ { 4 } r = \frac { 1 } { 2 } \log _ { 2 } r$$ (b)Solve the equation $$\log _ { 4 } \left( 5 x ^ { 2 } - 11 \right) = \log _ { 2 } ( 3 x - 5 )$$

\text { and } \mathbf { d } = \left( \begin{array} { c } - 4
2
- 11 \end{array} \right)$$

1. Find the position vector of \(E\) . The volume of a tetrahedron is given by the formula $$\text { volume } = \frac { 1 } { 3 } ( \text { area of base } ) \times ( \text { height } )$$
2. Find the volume of the tetrahedron \(A B C D\) . 4.(a)Given that \(x > 0 , y > 0 , x \neq 1\) and \(n > 0\) ,show that $$\log _ { x } y = \log _ { x ^ { n } } y ^ { n }$$
3. Solve the following,leaving your answers in the form \(2 ^ { p }\) ,where \(p\) is a rational number.
   1. \(\log _ { 2 } u + \log _ { 4 } u ^ { 2 } + \log _ { 8 } u ^ { 3 } + \log _ { 16 } u ^ { 4 } = 5\)
   2. \(\log _ { 2 } v + \log _ { 4 } v + \log _ { 8 } v + \log _ { 16 } v = 5\)
   3. \(\log _ { 4 } w ^ { 2 } + \frac { 3 \log _ { 8 } 64 } { \log _ { 2 } w } = 5\)

6.(i)Eden,who is confused about the laws of logarithms,states that $$\left( \log _ { 5 } p \right) ^ { 2 } = \log _ { 5 } \left( p ^ { 2 } \right)$$ and \(\log _ { 5 } ( q - p ) = \log _ { 5 } q - \log _ { 5 } p\) However,there is a value of \(p\) and a value of \(q\) for which both statements are correct.
Determine these values.
(ii)(a)Let \(r \in \mathbb { R } ^ { + } , r \neq 1\) .Prove that $$\log _ { r } A = \log _ { r ^ { 2 } } B \Rightarrow A ^ { 2 } = B$$ (b)Solve $$\log _ { 4 } \left( 3 x ^ { 3 } + 26 x ^ { 2 } + 40 x \right) = 2 + \log _ { 2 } ( x + 2 )$$

8 \includegraphics[max width=\textwidth, alt={}, center]{87012792-fa63-4003-875d-b8e7739037f1-4_489_697_274_667} The diagram shows the curves \(y = \log _ { 2 } x\) and \(y = \log _ { 2 } ( x - 3 )\).

1. Describe the geometrical transformation that transforms the curve \(y = \log _ { 2 } x\) to the curve \(y = \log _ { 2 } ( x - 3 )\).
2. The curve \(y = \log _ { 2 } x\) passes through the point ( \(a , 3\) ). State the value of \(a\).
3. The curve \(y = \log _ { 2 } ( x - 3 )\) passes through the point ( \(b , 1.8\) ). Find the value of \(b\), giving your answer correct to 3 significant figures.
4. The point \(P\) lies on \(y = \log _ { 2 } x\) and has an \(x\)-coordinate of \(c\). The point \(Q\) lies on \(y = \log _ { 2 } ( x - 3 )\) and also has an \(x\)-coordinate of \(c\). Given that the distance \(P Q\) is 4 units find the exact value of \(c\).

9

1. Sketch the graph of \(y = 4 k ^ { x }\), where \(k\) is a constant such that \(k > 1\). State the coordinates of any points of intersection with the axes.
2. The point \(P\) on the curve \(y = 4 k ^ { x }\) has its \(y\)-coordinate equal to \(20 k ^ { 2 }\). Show that the \(x\)-coordinate of \(P\) may be written as \(2 + \log _ { k } 5\).
3. (a) Use the trapezium rule, with two strips each of width \(\frac { 1 } { 2 }\), to find an expression for the approximate value of $$\int _ { 0 } ^ { 1 } 4 k ^ { x } \mathrm {~d} x$$ (b) Given that this approximate value is equal to 16 , find the value of \(k\).

8

1. Use logarithms to solve the equation \(5 ^ { 3 w - 1 } = 4 ^ { 250 }\), giving the value of \(w\) correct to 3 significant figures.
2. Given that \(\log _ { x } ( 5 y + 1 ) - \log _ { x } 3 = 4\), express \(y\) in terms of \(x\).

8 \includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-4_524_822_274_609} The diagram shows the curves \(y = a ^ { x }\) and \(y = 4 b ^ { x }\).

1. (a) State the coordinates of the point of intersection of \(y = a ^ { x }\) with the \(y\)-axis.
   (b) State the coordinates of the point of intersection of \(y = 4 b ^ { x }\) with the \(y\)-axis.
   (c) State a possible value for \(a\) and a possible value for \(b\).
2. It is now given that \(a b = 2\). Show that the \(x\)-coordinate of the point of intersection of \(y = a ^ { x }\) and \(y = 4 b ^ { x }\) can be written as $$x = \frac { 2 } { 2 \log _ { 2 } a - 1 } .$$

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. 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.

13. The function \(g\) is defined by $$\mathrm { g } ( x ) = \frac { 2 e ^ { x } - 5 } { e ^ { x } - 4 } \quad x \neq k , x > 0$$ where \(k\) is a constant.
a. Deduce the value of \(k\).
b. Prove that $$\mathrm { g } ^ { \prime } ( x ) < 0$$ For all values of \(x\) in the domain of g .
c. Find the range of values of \(a\) for which $$\mathrm { g } ( a ) > 0$$

1. (a) Given that

$$2 \log ( 4 - x ) = \log ( x + 8 )$$ show that $$x ^ { 2 } - 9 x + 8 = 0$$ (b) (i) Write down the roots of the equation $$x ^ { 2 } - 9 x + 8 = 0$$ (ii) State which of the roots in (b)(i) is not a solution of $$2 \log ( 4 - x ) = \log ( x + 8 )$$ giving a reason for your answer.

11 In this question you must show detailed reasoning.

1. A student is asked to solve the inequality \(x ^ { \frac { 1 } { 2 } } < 4\). The student argues that \(x ^ { \frac { 1 } { 2 } } < 4 \Leftrightarrow x < 16\), so that the solution is \(\{ x : x < 16 \}\).
   Comment on the validity of the student's argument.
2. Solve the inequality \(\left( \frac { 1 } { 2 } \right) ^ { x } < 4\).
3. Show that the equation \(2 \log _ { 2 } ( x + 8 ) - \log _ { 2 } ( x + 6 ) = 3\) has only one root.

5

1. Make \(y\) the subject of the formula \(\log _ { 10 } ( y - k ) = x \log _ { 10 } 2\), where \(k\) is a positive constant.
2. Sketch the graph of \(y\) against \(x\).

9 The function \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }\) is defined on the domain \(x \in \mathbb { R } , x \neq 0\).

1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
2. Write down the range of \(\mathrm { f } ^ { - 1 } ( x )\).

5

1. Given that $$\log _ { a } x = 3 \log _ { a } 6 - \log _ { a } 8$$ where \(a\) is a positive constant, show that \(x = 27\).
2. Write down the value of:
   1. \(\quad \log _ { 4 } 1\);
   2. \(\log _ { 4 } 4\);
   3. \(\log _ { 4 } 2\);
   4. \(\quad \log _ { 4 } 8\).

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 \(n\).

6

1. Sketch the curve with equation \(y = 2 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
2. 1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 2 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
   2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.
3. Describe a geometrical transformation that maps the graph of \(y = 2 ^ { x }\) onto the graph of \(y = 2 ^ { x + 7 } + 3\).
4. The curve \(y = 2 ^ { x + k } + 3\) intersects the \(y\)-axis at the point \(A ( 0,8 )\). Show that \(k = \log _ { m } n\), where \(m\) and \(n\) are integers.

8

1. Given that \(2 \log _ { k } x - \log _ { k } 5 = 1\), express \(k\) in terms of \(x\). Give your answer in a form not involving logarithms.
2. Given that \(\log _ { a } y = \frac { 3 } { 2 }\) and that \(\log _ { 4 } a = b + 2\), show that \(y = 2 ^ { p }\), where \(p\) is an expression in terms of \(b\).
   \includegraphics[max width=\textwidth, alt={}]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-09_2102_1717_605_150}

8

1. Given that \(\log _ { a } b = c\), express \(b\) in terms of \(a\) and \(c\).
2. By forming a quadratic equation, show that there is only one value of \(x\) which satisfies the equation \(2 \log _ { 2 } ( x + 7 ) - \log _ { 2 } ( x + 5 ) = 3\).