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

3

1. Show that the equation \(\log _ { 3 } ( 2 x + 1 ) = 1 + 2 \log _ { 3 } ( x - 1 )\) can be written as a quadratic equation in \(x\).
2. Hence solve the equation \(\log _ { 3 } ( 4 y + 1 ) = 1 + 2 \log _ { 3 } ( 2 y - 1 )\), giving your answer correct to 2 decimal places.

3 The variables \(x\) and \(y\) satisfy the relation \(2 ^ { y } = 3 ^ { 1 - 2 x }\).

1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. State the exact value of the gradient of this line.
2. Find the exact \(x\)-coordinate of the point of intersection of this line with the line \(y = 3 x\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.

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

1. Given

$$\frac { 3 ^ { x } } { 3 ^ { 4 y } } = 27 \sqrt { 3 }$$ find \(y\) as a simplified function of \(x\).

6. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations $$\begin{gathered} a b = 25 \\ \log _ { 4 } a - \log _ { 4 } b = 3 \end{gathered}$$ Show each step of your working, giving exact values for \(a\) and \(b\).

11. (i) Given that \(x\) is a positive real number, solve the equation $$\log _ { x } 324 = 4$$ writing your answer as a simplified surd.
(ii) Given that $$\log _ { a } ( 5 y - 4 ) - \log _ { a } ( 2 y ) = 3 \quad y > 0.8,0 < a < 1$$ express \(y\) in terms of \(a\).

1. (a) Show that the equation

$$2 \log _ { 2 } y = 5 - \log _ { 2 } x \quad x > 0 , y > 0$$ may be written in the form \(y ^ { 2 } = \frac { k } { x }\) where \(k\) is a constant to be found.
(b) Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} 2 \log _ { 2 } y = 5 - \log _ { 2 } x \\ \log _ { x } y = - 3 \end{gathered}$$ for \(x > 0 , y > 0\)

9. (a) Sketch the curve with equation $$y = 3 \times 4 ^ { x }$$ showing the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 6 ^ { 1 - x }\) meets the curve with equation \(y = 3 \times 4 ^ { x }\) at the point \(P\).
(b) Show that the \(x\) coordinate of \(P\) is \(\frac { \log _ { 10 } 2 } { \log _ { 10 } 24 }\)

3. (i) Solve $$7 ^ { x + 2 } = 3$$ giving your answer in the form \(x = \log _ { 7 } a\) where \(a\) is a rational number in its simplest form.
(ii) Using the laws of logarithms, solve $$1 + \log _ { 2 } y + \log _ { 2 } ( y + 4 ) = \log _ { 2 } ( 5 - y )$$

1. The weight of a baby mammal is monitored over a 16 -month period.

The weight of the mammal, \(w \mathrm {~kg}\), is given by $$w = \log _ { a } ( t + 5 ) - \log _ { a } 4 \quad 2 \leqslant t \leqslant 18$$ where \(t\) is the age of the mammal in months and \(a\) is a constant.
Given that the weight of the mammal was 10 kg when \(t = 3\)

1. show that \(a = 1.072\) correct to 3 decimal places. Using \(a = 1.072\)
2. find an equation for \(t\) in terms of \(w\)
3. find the value of \(t\) when \(w = 15\), giving your answer to 3 significant figures.

5. (a) Find the positive value of \(x\) such that $$\log _ { x } 64 = 2$$ (b) Solve for \(x\) $$\log _ { 2 } ( 11 - 6 x ) = 2 \log _ { 2 } ( x - 1 ) + 3$$

8. (a) Find the value of \(y\) such that $$\log _ { 2 } y = - 3$$ (b) Find the values of \(x\) such that $$\frac { \log _ { 2 } 32 + \log _ { 2 } 16 } { \log _ { 2 } x } = \log _ { 2 } x$$

3. Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which

1. \(5 ^ { x } = 10\),
2. \(\log _ { 3 } ( x - 2 ) = - 1\).

8. A dose of antibiotics is given to a patient. The amount of the antibiotic, \(x\) milligrams, in the patient's bloodstream \(t\) hours after the dose was given, is found to satisfy the equation $$\log _ { 10 } x = 2.74 - 0.079 t$$

1. Show that this equation can be written in the form $$x = p q ^ { - t }$$ where \(p\) and \(q\) are constants to be found. Give the value of \(p\) to the nearest whole number and the value of \(q\) to 2 significant figures.
2. With reference to the equation in part (a), interpret the value of the constant \(p\). When a different dose of the antibiotic is given to another patient, the values of \(x\) and \(t\) satisfy the equation $$x = 400 \times 1.4 ^ { - t }$$
3. Use calculus to find, to 2 significant figures, the value of \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when \(t = 5\)

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 The mass, \(M \mathrm {~kg}\), of a species of tree can be modelled by the equation $$\log _ { 10 } M = 1.93 \log _ { 10 } r + 0.684$$ where \(r \mathrm {~cm}\) is the base radius of the tree.
The base radius of a particular tree of this species is 45 cm .
According to the model,

1. find the mass of this tree, giving your answer to 2 significant figures.
2. Show that the equation of the model can be written in the form $$M = p r ^ { q }$$ giving the values of the constants \(p\) and \(q\) to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(p\). Q

4. $$y = \log _ { 10 } ( 2 x + 1 )$$

1. Express \(x\) in terms of \(y\).
2. Hence, giving your answer in terms of \(x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) 4.
   1. Express \(x\) in terms of \(y\).

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      (2)
   2. Hence, giving your answer in terms of \(x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-10_2662_111_107_1950}

1. The function \(f\) is defined by

$$\mathrm { f } : x \mapsto \ln ( 4 - 2 x ) , \quad x < 2 \quad \text { and } \quad x \in \mathbb { R } .$$

1. Show that the inverse function of f is defined by $$\mathrm { f } ^ { - 1 } : x \mapsto 2 - \frac { 1 } { 2 } \mathrm { e } ^ { x }$$ and write down the domain of \(\mathrm { f } ^ { - 1 }\).
2. Write down the range of \(\mathrm { f } ^ { - 1 }\).
3. In the space provided on page 16, sketch the graph of \(y = f ^ { - 1 } ( x )\). State the coordinates of the points of intersection with the \(x\) and \(y\) axes. The graph of \(y = x + 2\) crosses the graph of \(y = f ^ { - 1 } ( x )\) at \(x = k\). The iterative formula $$x _ { n + 1 } = - \frac { 1 } { 2 } e ^ { x _ { n } } , x _ { 0 } = - 0.3$$ is used to find an approximate value for \(k\).
4. Calculate the values of \(x _ { 1 }\) and \(x _ { 2 }\), giving your answers to 4 decimal places.
5. Find the value of \(k\) to 3 decimal places.

8. The function \(f\) is defined by $$\mathrm { f } : x \rightarrow 3 - 2 \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$

1. Find the inverse function, \(\mathrm { f } ^ { - 1 } ( x )\) and give its domain.
2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \ln x\). The equation \(\mathrm { f } ( t ) = k \mathrm { e } ^ { t }\), where \(k\) is a positive constant, has exactly one real solution.
3. Find the value of \(k\).

8

1. On a single diagram, sketch the curves with the following equations. In each case state the coordinates of any points of intersection with the axes.
   1. \(y = a ^ { x }\), where \(a\) is a constant such that \(a > 1\).
   2. \(y = 2 b ^ { x }\), where \(b\) is a constant such that \(0 < b < 1\).
   3. The curves in part (i) intersect at the point \(P\). Prove that the \(x\)-coordinate of \(P\) is $$\frac { 1 } { \log _ { 2 } a - \log _ { 2 } b } .$$

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)

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 \(\log _ { a } x\).

6

1. Write down the values of \(\log _ { a } 1\) and \(\log _ { a } a\), where \(a > 1\).
2. Show that \(\log _ { a } x ^ { 10 } - 2 \log _ { a } \left( \frac { x ^ { 3 } } { 4 } \right) = 4 \log _ { a } ( 2 x )\).