1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form

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.

9 A biologist is investigating the growth of bacteria in a piece of bread.
He believes that the number, \(N\), of bacteria after \(t\) hours may be modelled by the relationship \(N = A \times 2 ^ { k t }\), where \(A\) and \(k\) are constants.

1. Show that, according to the model, the graph of \(\log _ { 10 } N\) against \(t\) is a straight line. Give, in terms of \(A\) and \(k\),
   The biologist measures the number of bacteria at regular intervals over 22 hours and plots a graph of \(\log _ { 10 } N\) against \(t\). He finds that the graph is approximately a straight line with gradient 0.20 . The line crosses the vertical axis at 2.0 .
2. Find the values of \(A\) and \(k\).
3. Use the model to predict the number of bacteria after 24 hours.
4. Give a reason why the model may not be appropriate for large values of \(t\).

9 In 2012 Adam bought a second hand car for \(\pounds 8500\). Each year Adam has his car valued. He believes that there is a non-linear relationship between \(t\), the time in years since he bought the car, and \(V\), the value of the car in pounds. Fig. 9.1 shows successive values of \(V\) and \(\log _ { 10 } V\). \begin{table}[h] \end{table} Adam uses a spreadsheet to plot the points ( \(t , \log _ { 10 } V\) ) shown in Fig. 9.1, and then generates a line of best fit for these points. The line passes through the points \(( 0,3.93 )\) and \(( 4,3.58 )\). A copy of his graph is shown in Fig. 9.2. \begin{figure}[h] \end{figure}

1. Find an expression for \(\log _ { 10 } V\) in terms of \(t\).
2. Find a model for \(V\) in the form \(V = A \times b ^ { t }\), where \(A\) and \(b\) are constants to be determined. Give the values of \(A\) and \(b\) correct to 2 significant figures. In 2017 Adam's car was valued at \(\pounds 3150\).
3. Determine whether the model is a good fit for this data. A company called Webuyoldcars pays \(\pounds 500\) for any second hand car. Adam decides that he will sell his car to this company when the annual valuation of his car is less than \(\pounds 500\).
4. According to the model, after how many years will Adam sell his car to Webuyoldcars?

3 You are given that \(y = A e ^ { 0.02 t }\).

• Make \(t\) the subject of the formula.
• Find the value of \(t\) when \(y = 10 ^ { 8 }\) and \(A = 6.62 \times 10 ^ { 7 }\).

14 Alex places a hot object into iced water and records the temperature \(\theta ^ { \circ } \mathrm { C }\) of the object every minute. The temperature of an object \(t\) minutes after being placed in iced water is modelled by \(\theta = \theta _ { 0 } \mathrm { e } ^ { - k t }\) where \(\theta _ { 0 }\) and \(k\) are constants whose values depend on the characteristics of the object. The temperature of Alex's object is \(82 ^ { \circ } \mathrm { C }\) when it is placed into the water. After 5 minutes the temperature is \(27 ^ { \circ } \mathrm { C }\).

1. Find the values of \(\theta _ { 0 }\) and \(k\) that best model the data.
2. Explain why the model may not be suitable in the long term if Alex does not top up the ice in the water.
3. Show that the model with the values found in part (a) can be written as \(\ln \theta = \mathrm { a } -\) bt where \(a\) and \(b\) are constants to be determined. Ben places a different object into iced water at the same time as Alex. The model for Ben's object is \(\ln \theta = 3.4 - 0.08 t\).
4. Determine each of the following:
   Find this time and the corresponding temperature.

5 A social media website launched on 1 January 2017. The owners of the website report the number of users the site has at the start of each month. They believe that the relationship between the number of users, \(n\), and the number of months after launch, \(t\), can be modelled by \(n = a \times 2 ^ { k t }\) where \(a\) and \(k\) are constants.

1. Show that, according to the model, the graph of \(\log _ { 10 } n\) against \(t\) is a straight line.
2. Fig. 5 shows a plot of the values of \(t\) and \(\log _ { 10 } n\) for the first seven months. The point at \(t = 1\) is for 1 February 2017, and so on. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-6_831_1442_609_388} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure} Find estimates of the values of \(a\) and \(k\).
3. The owners of the website wanted to know the date on which they would report that the website had half a million users. Use the model to estimate this date.
4. Give a reason why the model may not be appropriate for large values of \(t\).

6

1. 1. Write down the derivative of \(\mathrm { e } ^ { \mathrm { kx } }\), where \(k\) is a constant.
   2. A business has been running since 2009. They sell maths revision resources online. Give a reason why an exponential growth model might be suitable for the annual profits for the business. Fig. 6 shows the relationship between the annual profits of the business in thousands of pounds ( \(y\) ) and the time in years after \(2009 ( x )\). The graph of lny plotted against \(x\) is approximately a straight line. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-07_1052_1157_751_242} \captionsetup{labelformat=empty} \caption{Fig. 6}
      \end{figure}
2. Show that the straight line is consistent with a model of the form \(\mathbf { y } = \mathrm { Ae } ^ { \mathrm { kx } }\), where \(A\) and \(k\) are constants.
3. Estimate the values of \(A\) and \(k\).
4. Use the model to predict the profit in the year 2020.
5. How reliable do you expect the prediction in part (d) to be? Justify your answer.

5

1. The diagram shows part of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the point \(( a , 0 )\) and the \(y\)-axis at the point \(( 0 , - b )\). \includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-4_569_853_1206_589} On separate diagrams, sketch the curves with the following equations. On each diagram, indicate, in terms of \(a\) or \(b\), the coordinates of the points where the curve crosses the coordinate axes.
   1. \(y = | \mathrm { f } ( x ) |\).
   2. \(\quad y = 2 \mathrm { f } ( x )\).
2. 1. Describe a sequence of geometrical transformations that maps the graph of \(y = \ln x\) onto the graph of \(y = 4 \ln ( x + 1 ) - 2\).
   2. Find the exact values of the coordinates of the points where the graph of \(y = 4 \ln ( x + 1 ) - 2\) crosses the coordinate axes.

6. A radioactive isotope decays in such a way that the rate of change of the number \(N\) of radioactive atoms present after \(t\) days, is proportional to \(N\).

1. Write down a differential equation relating \(N\) and \(t\).
2. Show that the general solution may be written as \(N = A \mathrm { e } ^ { - k t }\), where \(A\) and \(k\) are positive constants. Initially the number of radioactive atoms present is \(7 \times 10 ^ { 18 }\) and 8 days later the number present is \(3 \times 10 ^ { 17 }\).
3. Find the value of \(k\).
4. Find the number of radioactive atoms present after a further 8 days.

6 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.]
The variables \(x\) and \(y\) are known to be related by an equation of the form $$y = k x ^ { n }$$ where \(k\) and \(n\) are constants.
Experimental evidence has provided the following approximate values:

1. Complete the table in Figure 1, showing values of \(X\) and \(Y\), where $$X = \log _ { 10 } x \quad \text { and } \quad Y = \log _ { 10 } y$$ Give each value to two decimal places.
2. Show that if \(y = k x ^ { n }\), then \(X\) and \(Y\) must satisfy an equation of the form $$Y = a X + b$$
3. Draw on Figure 2 a linear graph relating \(X\) and \(Y\).
4. Find an estimate for the value of \(n\).

4 The variables \(x\) and \(y\) are related by an equation of the form $$y = a x ^ { b }$$ where \(a\) and \(b\) are constants.

1. Using logarithms to base 10 , reduce the relation \(y = a x ^ { b }\) to a linear law connecting \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
2. The diagram shows the linear graph that results from plotting \(\log _ { 10 } y\) against \(\log _ { 10 } x\). \includegraphics[max width=\textwidth, alt={}, center]{49539feb-f842-49f4-b809-72e8147072e7-3_711_1223_1503_411} Find the values of \(a\) and \(b\).

6

1. The matrix \(\mathbf { X }\) is defined by \(\left[ \begin{array} { l l } 1 & 2 \\ 3 & 0 \end{array} \right]\).
   1. Given that \(\mathbf { X } ^ { 2 } = \left[ \begin{array} { c c } m & 2 \\ 3 & 6 \end{array} \right]\), find the value of \(m\).
   2. Show that \(\mathbf { X } ^ { 3 } - 7 \mathbf { X } = n \mathbf { I }\), where \(n\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
2. It is given that \(\mathbf { A } = \left[ \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right]\).
   1. Describe the geometrical transformation represented by \(\mathbf { A }\).
   2. The matrix \(\mathbf { B }\) represents an anticlockwise rotation through \(45 ^ { \circ }\) about the origin. Show that \(\mathbf { B } = k \left[ \begin{array} { r r } 1 & - 1 \\ 1 & 1 \end{array} \right]\), where \(k\) is a surd.
   3. Find the image of the point \(P ( - 1,2 )\) under an anticlockwise rotation through \(45 ^ { \circ }\) about the origin, followed by the transformation represented by \(\mathbf { A }\). \(7 \quad\) The variables \(y\) and \(x\) are related by an equation of the form $$y = a x ^ { n }$$ where \(a\) and \(n\) are constants. Let \(Y = \log _ { 10 } y\) and \(X = \log _ { 10 } x\).

5 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.] The variables \(x\) and \(y\) are known to be related by an equation of the form $$y = a b ^ { x }$$ where \(a\) and \(b\) are constants. The following approximate values of \(x\) and \(y\) have been found.

1. Complete the table in Figure 1, showing values of \(x\) and \(Y\), where \(Y = \log _ { 10 } y\). Give each value of \(Y\) to three decimal places.
2. Show that, if \(y = a b ^ { x }\), then \(x\) and \(Y\) must satisfy an equation of the form $$Y = m x + c$$
3. Draw on Figure 2 a linear graph relating \(x\) and \(Y\).
4. Hence find estimates for the values of \(a\) and \(b\).

4 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.]
The variables \(x\) and \(y\) are related by an equation of the form $$y = a x + \frac { b } { x + 2 }$$ where \(a\) and \(b\) are constants.

1. The variables \(X\) and \(Y\) are defined by \(X = x ( x + 2 ) , Y = y ( x + 2 )\). Show that \(Y = a X + b\).
2. The following approximate values of \(x\) and \(y\) have been found:

   \(x\)	1	2	3	4
   \(y\)	0.40	1.43	2.40	3.35

   1. Complete the table in Figure 1, showing values of \(X\) and \(Y\).
   2. Draw on Figure 2 a linear graph relating \(X\) and \(Y\).
   3. Estimate the values of \(a\) and \(b\).

4 The variables \(x\) and \(y\) are known to be related by an equation of the form $$y = a b ^ { x }$$ where \(a\) and \(b\) are constants.

1. Given that \(Y = \log _ { 10 } y\), show that \(x\) and \(Y\) must satisfy an equation of the form $$Y = m x + c$$
2. The diagram shows the linear graph which has equation \(Y = m x + c\). \includegraphics[max width=\textwidth, alt={}, center]{932d4c7e-6514-4543-b1d1-753fca5a08fd-5_744_720_833_699} Use this graph to calculate:
   1. an approximate value of \(y\) when \(x = 2.3\), giving your answer to one decimal place;
   2. an approximate value of \(x\) when \(y = 80\), giving your answer to one decimal place.
      (You are not required to find the values of \(m\) and \(c\).)

4 The variables \(x\) and \(Y\), where \(Y = \log _ { 10 } y\), are related by the equation $$Y = m x + c$$ where \(m\) and \(c\) are constants.

1. Given that \(y = a b ^ { x }\), express \(a\) in terms of \(c\), and \(b\) in terms of \(m\).
2. It is given that \(y = 12\) when \(x = 1\) and that \(y = 27\) when \(x = 5\). On the diagram below, draw a linear graph relating \(x\) and \(Y\).
3. Use your graph to estimate, to two significant figures:
   1. the value of \(y\) when \(x = 3\);
   2. the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{7441c4e6-5448-483b-b100-f8076e7e6cd8-3_976_1173_1110_484}

13 Answer part (ii) of this question on the insert provided. The table gives a firm's monthly profits for the first few months after the start of its business, rounded to the nearest \(\pounds 100\). The firm's profits, \(\pounds y\), for the \(x\) th month after start-up are modelled by $$y = k \times 10 ^ { a x }$$ where \(a\) and \(k\) are constants.

1. Show that, according to this model, a graph of \(\log _ { 10 } y\) against \(x\) gives a straight line of gradient \(a\) and intercept \(\log _ { 10 } k\).
2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(x\), drawing by eye a line of best fit.
3. Use your graph to find an equation for \(y\) in terms of \(x\) for this model.
4. For which month after start-up does this model predict profits of about \(\pounds 75000\) ?
5. State one way in which this model is unrealistic.

12 Answer part (ii) of this question on the insert provided. The proposal for a major building project was accepted, but actual construction was delayed. Each year a new estimate of the cost was made. The table shows the estimated cost, \(\pounds y\) million, of the project \(t\) years after the project was first accepted. The relationship between \(y\) and \(t\) is modelled by \(y = a b ^ { t }\), where \(a\) and \(b\) are constants.

1. Show that \(y = a b ^ { t }\) may be written as $$\log _ { 10 } y = \log _ { 10 } a + t \log _ { 10 } b$$
2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(t\), drawing by eye a line of best fit.
3. Use your graph and the results of part (i) to find the values of \(\log _ { 10 } a\) and \(\log _ { 10 } b\) and hence \(a\) and \(b\).
4. According to this model, what was the estimated cost of the project when it was first accepted?
5. Find the value of \(t\) given by this model when the estimated cost is \(\pounds 1000\) million. Give your answer rounded to 1 decimal place.

9 Answer parts (ii) and (iii) of this question on the Insert provided. The bat population of a colony is being investigated and data are collected of the estimated number of bats in the colony at the beginning of each year. It is thought that the population may be modelled by the formula $$P = P _ { 0 } \mathrm { e } ^ { k t }$$ where \(P _ { 0 }\) and \(k\) are constants, \(P\) is the number of bats and \(t\) is the number of years after the start of the collection of data.

1. Explain why a graph of \(\ln P\) against \(t\) should give a straight line. State the gradient and intercept of this line.
2. The data collected are as follows.

   Time \(( t\) years \()\)	0	1	2	3	4
   Number of bats, \(P\)	100	170	300	340	360

   Using the first three pairs of data in the table, plot \(\ln P\) against \(t\) on the axes given on the Insert, and hence estimate values for \(P _ { 0 }\) and \(k\).
   (Work to three significant figures.) This model assumes exponential growth, and assumes that once born a bat does not die, continuing to reproduce. This is unrealistic and so a second model is proposed with formula $$P = 150 \arctan ( t - 1 ) + 170$$ (You are reminded that arctan values should be given in radians.)
3. Plot on a single graph on the Insert the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) for your values of \(P _ { 0 }\) and \(k\) and \(P = 150 \arctan ( t - 1 ) + 170\). The data pairs in the table above have been plotted for you.
4. Using the second model calculate an estimate of the number of years it is before the bat population exceeds 375. \section*{Insert for question 3.}
5. Sketch the graph of \(y = 2 \mathrm { f } ( x )\) \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_641_1431_541_354}
6. Sketch the graph of \(y = \mathrm { f } ( 2 x )\). \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_691_1539_1468_374} \section*{Insert for question 9.}
7. Plot \(\ln P\) against \(t\). \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_704_1442_443_338}
8. Plot the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) and \(P = 150 \arctan ( t - 1 ) + 170\) for your values of \(P _ { 0 }\) and \(k\). The data pairs are plotted on the graph. \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_780_1399_1546_333}

7 [Figure 1, printed on the insert, is provided for use in this question.]
The variables \(x\) and \(y\) are known to be related by an equation of the form $$y ^ { 3 } = a x ^ { 2 } + b$$ where \(a\) and \(b\) are constants. Experimental evidence has provided the following approximate values:

1. On Figure 1, draw a linear graph connecting the variables \(X\) and \(Y\), where $$X = x ^ { 2 } \quad \text { and } \quad Y = y ^ { 3 }$$
2. From your graph, find approximate values for the constants \(a\) and \(b\).

10 As part of an experiment, Zena puts a bucket of hot water outside on a day when the outside temperature is \(0 ^ { \circ } \mathrm { C }\). She measures the temperature of the water after 10 minutes and after 20 minutes. Her results are shown below. Zena models the relationship between \(\theta\), the temperature of the water in \({ } ^ { \circ } \mathrm { C }\), and \(t\), the time in minutes, by $$\theta = A \times 10 ^ { - k t }$$ where \(A\) and \(k\) are constants. 10

1. Using \(t = 0\), explain how the value of \(A\) relates to the experiment. 10
2. Show that $$\log _ { 10 } \theta = \log _ { 10 } A - k t$$ 10
3. Using Zena's results, calculate the values of \(A\) and \(k\).
   10
4. Zena states that the temperature of the water will be less than \(1 ^ { \circ } \mathrm { C }\) after 45 minutes. Determine whether the model supports this statement.
   10
5. Explain why Zena's model is unlikely to accurately give the value of \(\theta\) after 45 minutes.

9 The table below shows the annual global production of plastics, \(P\), measured in millions of tonnes per year, for six selected years. It is thought that \(P\) can be modelled by $$P = A \times 10 ^ { k t }$$ where \(t\) is the number of years after 1980 and \(A\) and \(k\) are constants.
9

1. Show algebraically that the graph of \(\log _ { 10 } P\) against \(t\) should be linear.
   9
2. (i) Complete the table below.

   \(\boldsymbol { t }\)	0	5	10	15	20	25
   \(\boldsymbol { \operatorname { l o g } } _ { \mathbf { 1 0 } } \boldsymbol { P }\)	1.88	1.97	2.08		2.31

   9 (b) (ii) Plot \(\log _ { 10 } P\) against \(t\), and draw a line of best fit for the data. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-13_1203_1308_360_367} 9
3. (i) Hence, show that \(k\) is approximately 0.02
   9 (c) (ii) Find the value of \(A\).
   9
4. Using the model with \(k = 0.02\) predict the number of tonnes of annual global production of plastics in 2030. 9
5. Using the model with \(k = 0.02\) predict the year in which \(P\) first exceeds 8000
   9
6. Give a reason why it may be inappropriate to use the model to make predictions about future annual global production of plastics. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-15_2488_1716_219_153}

6 Victoria, a market researcher, believes the average weekly value, \(\pounds V\) million, of online grocery sales in the UK has grown exponentially since 2009. Victoria models the incomplete data, shown in the table, using the formula $$V = a \times b ^ { N }$$ where \(N\) is the number of years since 2009 and \(a\) and \(b\) are constants. 6

1. Victoria wishes to determine the values of \(a\) and \(b\) in her formula.
   To do this she plots a graph of \(\log _ { 10 } V\) against \(N\) and then draws a line of best fit as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-08_757_1040_1169_589} The equation of Victoria's line of best fit is $$\log _ { 10 } V = 0.057 N + 1.76$$ 6
   1. (i) Use the equation of Victoria's line of best fit to show that, correct to three significant figures, \(a = 57.5\) [0pt] [1 mark]
      6
   2. (ii) Use the equation of Victoria's line of best fit to find the value of \(b\) Give your answer to three significant figures. 6
   3. According to Victoria's model, state the yearly percentage increase in the average weekly value of online grocery sales. 6
   4. 1. Use Victoria's model to predict the average weekly value of online grocery sales in 2025.
         6
   5. (ii) Explain why the prediction made in part (c)(i) may be unreliable.