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

10 A function \(y = \mathrm { f } ( x )\) may be modelled by the equation \(y = a x ^ { b }\).

1. Show why, if this is so, then plotting \(\log y\) against \(\log x\) will produce a straight line graph. Explain how \(a\) and \(b\) may be determined experimentally from the graph.
2. Values of \(x\) and \(y\) are given below. By plotting a graph of logy against log \(x\), show that the model above is appropriate for this set of data and find values of \(a\) and \(b\) given that \(a\) is an integer and \(b\) can be written as a fraction with a denominator less than 10 .

   \(x\)	2	3	4	5	6
   \(y\)	4.6	5.0	5.3	5.5	5.7

3. Use your formula from part (ii) to estimate the value of \(y\) when \(x = 2.8\).

10 A culture of bacteria is observed during an experiment. The number of bacteria is denoted by \(N\) and the time in hours after the start of the experiment by \(t\).
The table gives observations of \(t\) and \(N\).

1. Plot the points \(( t , N )\) on graph paper and join them with a smooth curve.
2. Explain why the curve suggests why the relationship connecting \(t\) and \(N\) may be of the form \(N = a b ^ { t }\).
3. Explain how, by using logarithms, the curve given by plotting \(N\) against \(t\) can be transformed into a straight line.
   State the gradient of this straight line and its intercept with the vertical axis in terms of \(a\) and \(b\).
4. Complete a table of values for \(\log _ { 10 } N\) and plot the points \(\left( t , \log _ { 10 } N \right)\) on graph paper. Draw the best fit line through the points and use it to estimate the values of \(a\) and \(b\).

8. The population in thousands, \(P\), of a town at time \(t\) years after \(1 ^ { \text {st } }\) January 1980 is modelled by the formula $$P = 30 + 50 \mathrm { e } ^ { 0.002 t }$$ Use this model to estimate

1. the population of the town on \(1 ^ { \text {st } }\) January 2010,
2. the year in which the population first exceeds 84000 . The population in thousands, \(Q\), of another town is modelled by the formula $$Q = 26 + 50 \mathrm { e } ^ { 0.003 t }$$
3. Show that the value of \(t\) when \(P = Q\) is a solution of the equation $$t = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t } \right)$$
4. Use the iterative formula $$t _ { n + 1 } = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t _ { n } } \right)$$ with \(t _ { 0 } = 50\) to find \(t _ { 1 } , t _ { 2 }\) and \(t _ { 3 }\) and hence, the year in which the populations of these two towns will be equal according to these models.

6

1. \(t\)	0	10	20
   \(X\)	275	440

   The quantity \(X\) is increasing exponentially with respect to time \(t\). The table above shows values of \(X\) for different values of \(t\). Find the value of \(X\) when \(t = 20\).
2. The quantity \(Y\) is decreasing exponentially with respect to time \(t\) where $$Y = 80 \mathrm { e } ^ { - 0.02 t }$$
   1. Find the value of \(t\) for which \(Y = 20\), giving your answer correct to 2 significant figures.
   2. Find by differentiation the rate at which \(Y\) is decreasing when \(t = 30\), giving your answer correct to 2 significant figures.

12 The table shows the size of a population of house sparrows from 1980 to 2005. The 'red alert' category for birds is used when a population has decreased by at least \(50 \%\) in the previous 25 years.

1. Show that the information for this population is consistent with the house sparrow being on red alert in 2005. The size of the population may be modelled by a function of the form \(P = a \times 10 ^ { - k t }\), where \(P\) is the population, \(t\) is the number of years after 1980, and \(a\) and \(k\) are constants.
2. Write the equation \(P = a \times 10 ^ { - k t }\) in logarithmic form using base 10, giving your answer as simply as possible.
3. Complete the table and draw the graph of \(\log _ { 10 } P\) against \(t\), drawing a line of best fit by eye.
4. Use your graph to find the values of \(a\) and \(k\) and hence the equation for \(P\) in terms of \(t\).
5. Find the size of the population in 2015 as predicted by this model. Would the house sparrow still be on red alert? Give a reason for your answer.

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

8 Fig. 8 shows the graph of \(\log _ { 10 } y\) against \(\log _ { 10 } x\). It is a straight line passing through the points \(( 2,8 )\) and \(( 0,2 )\). \begin{figure}[h] \end{figure} Find the equation relating \(\log _ { 10 } y\) and \(\log _ { 10 } x\) and hence find the equation relating \(y\) and \(x\).

7 A particle is moving vertically downwards in a liquid. Initially its velocity is zero, and after \(t\) seconds it is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Its terminal (long-term) velocity is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A model of the particle's motion is proposed. In this model, \(v = 5 \left( 1 - \mathrm { e } ^ { - 2 t } \right)\).

1. Show that this equation is consistent with the initial and terminal velocities. Calculate the velocity after 0.5 seconds as given by this model.
2. Verify that \(v\) satisfies the differential equation \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 - 2 v\). In a second model, \(v\) satisfies the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 - 0.4 v ^ { 2 }$$ As before, when \(t = 0 , v = 0\).
3. Show that this differential equation may be written as $$\frac { 10 } { ( 5 - v ) ( 5 + v ) } \frac { \mathrm { d } v } { \mathrm {~d} t } = 4$$ Using partial fractions, solve this differential equation to show that $$t = \frac { 1 } { 4 } \ln \left( \frac { 5 + v } { 5 - v } \right)$$ This can be re-arranged to give \(v = \frac { 5 \left( 1 - \mathrm { e } ^ { - 4 t } \right) } { 1 + \mathrm { e } ^ { - 4 t } }\). [You are not required to show this result.]
4. Verify that this model also gives a terminal velocity of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the velocity after 0.5 seconds as given by this model. The velocity of the particle after 0.5 seconds is measured as \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
5. Which of the two models fits the data better?

7

1. Show that the substitution \(x = \mathrm { e } ^ { t }\) transforms the differential equation $$\text { into } \quad \begin{aligned} x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y & = 3 + 20 \sin ( \ln x ) \\ \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 6 y & = 3 + 20 \sin t \end{aligned}$$ (7 marks)
2. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 6 y = 3 + 20 \sin t$$ (11 marks)
3. Write down the general solution of the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y = 3 + 20 \sin ( \ln x )$$

6 A mobile phone company records their annual sales on \(31 ^ { \text {st } }\) December every year.
Paul thinks that the annual sales, \(S\) million, can be modelled by the equation \(S = a b ^ { t }\), where \(a\) and \(b\) are both positive constants and \(t\) is the number of years since \(31 ^ { \text {st } }\) December 2015. Paul tests his theory by using the annual sales figures from \(31 ^ { \text {st } }\) December 2015 to \(31 { } ^ { \text {st } }\) December 2019. He plots these results on a graph, with \(t\) on the horizontal axis and \(\log _ { 10 } S\) on the vertical axis.

1. Explain why, if Paul's model is correct, the results should lie on a straight line of best fit on his graph. The results lie on a straight line of best fit which has a gradient of 0.146 and an intercept on the vertical axis of 0.583 .
2. Use these values to obtain estimates for \(a\) and \(b\), correct to 2 significant figures.
3. Use this model to predict the year in which, on the \(31 { } ^ { \text {st } }\) December, the annual sales would first be recorded as greater than 200 million.
4. Give a reason why this prediction may not be reliable.

8

1. Substance \(A\) is decaying exponentially such that its mass is \(m\) grams at time \(t\) minutes. Find the missing values of \(m\) and \(t\) in the following table.

   \(t\)	0	10		50
   \(m\)	1250	750	450

2. Substance \(B\) is also decaying exponentially, according to the model \(m = 160 \mathrm { e } ^ { - 0.055 t }\), where \(m\) grams is its mass after \(t\) minutes.
   1. Determine the value of \(t\) for which the mass of substance \(B\) is half of its original mass.
   2. Determine the rate of decay of substance \(B\) when \(t = 15\).
3. State whether substance \(A\) or substance \(B\) is decaying at a faster rate, giving a reason for your answer.

11 The owners of an online shop believe that their sales can be modelled by \(S = a b ^ { t }\), where \(a\) and \(b\) are both positive constants, \(S\) is the number of items sold in a month and \(t\) is the number of complete months since starting their online shop. The sales for the first six months are recorded, and the values of \(\log _ { 10 } S\) are plotted against \(t\) in the graph below. The graph is repeated in the Printed Answer Booklet. \includegraphics[max width=\textwidth, alt={}, center]{9473b8f7-616a-485e-963b-696c6640ae6b-08_1203_1408_552_244}

1. Explain why the graph suggests that the given model is appropriate. The owners believe that \(a = 120\) and \(b = 1.15\) are good estimates for the parameters in the model.
2. Show that the graph supports these estimates for the parameters.
3. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of items sold in the seventh month after opening.
4. 1. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of months after opening when the total number of items sold after opening will first exceed 70000 .
   2. Comment on how reliable this prediction may be.

13. \begin{figure}[h] \end{figure} The value of a rare painting, \(\pounds V\), is modelled by the equation \(V = p q ^ { t }\), where \(p\) and \(q\) are constants and \(t\) is the number of years since the value of the painting was first recorded on 1st January 1980. The line \(l\) shown in Figure 3 illustrates the linear relationship between \(t\) and \(\log _ { 10 } V\) since 1st January 1980. The equation of line \(l\) is \(\log _ { 10 } V = 0.05 t + 4.8\)

1. Find, to 4 significant figures, the value of \(p\) and the value of \(q\).
2. With reference to the model interpret
   1. the value of the constant \(p\),
   2. the value of the constant \(q\).
3. Find the value of the painting, as predicted by the model, on 1st January 2010, giving your answer to the nearest hundred thousand pounds.

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

Solutions relying on calculator technology are not acceptable.

1. Solve the equation $$x \sqrt { 2 } - \sqrt { 18 } = x$$ writing the answer as a surd in simplest form.
2. Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$

7. \begin{figure}[h] \end{figure} A chimney emits smoke particles.
On a particular day, the concentration of smoke particles in the air emitted by this chimney, \(P\) parts per million, is measured at various distances, \(x \mathrm {~km}\), from the chimney. Figure 2 shows a sketch of the linear relationship between \(\log _ { 10 } P\) and \(x\) that is used to model this situation. The line passes through the point ( \(0,3.3\) ) and the point ( \(6,2.1\) )

1. Find a complete equation for the model in the form $$P = a b ^ { x }$$ where \(a\) and \(b\) are constants. Give the value of \(a\) and the value of \(b\) each to 4 significant figures.
2. With reference to the model, interpret the value of \(a b\)

13. \begin{figure}[h] \end{figure} The resting heart rate, \(h\), of a mammal, measured in beats per minute, is modelled by the equation $$h = p m ^ { q }$$ where \(p\) and \(q\) are constants and \(m\) is the mass of the mammal measured in kg .
Figure 2 illustrates the linear relationship between \(\log _ { 10 } h\) and \(\log _ { 10 } m\) The line meets the vertical \(\log _ { 10 } h\) axis at 2.25 and has a gradient of - 0.235

1. Find, to 3 significant figures, the value of \(p\) and the value of \(q\). A particular mammal has a mass of 5 kg and a resting heart rate of 119 beats per minute.
2. Comment on the suitability of the model for this mammal.
3. With reference to the model, interpret the value of the constant \(p\).

11. \begin{figure}[h] \end{figure} The value, \(V\) pounds, of a mobile phone, \(t\) months after it was bought, is modelled by $$V = a b ^ { t }$$ where \(a\) and \(b\) are constants.
Figure 2 shows the linear relationship between \(\log _ { 10 } V\) and \(t\).
The line passes through the points \(( 0,3 )\) and \(( 10,2.79 )\) as shown.
Using these points,

1. find the initial value of the phone,
2. find a complete equation for \(V\) in terms of \(t\), giving the exact value of \(a\) and giving the value of \(b\) to 3 significant figures. Exactly 2 years after it was bought, the value of the phone was \(\pounds 320\)
3. Use this information to evaluate the reliability of the model.

1. In a controlled experiment, the number of microbes, \(N\), present in a culture \(T\) days after the start of the experiment were counted. \(N\) and \(T\) are expected to satisfy a relationship of the form

$$N = a T ^ { b } , \quad \text { where } a \text { and } b \text { are constants }$$

1. Show that this relationship can be expressed in the form $$\log _ { 10 } N = m \log _ { 10 } T + c$$ giving \(m\) and \(c\) in terms of the constants \(a\) and/or \(b\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-18_1232_1046_804_513} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows the line of best fit for values of \(\log _ { 10 } N\) plotted against values of \(\log _ { 10 } T\)
2. Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.
3. Explain why the information provided could not reliably be used to estimate the day when the number of microbes in the culture first exceeds 1000000 .
4. With reference to the model, interpret the value of the constant \(a\).

1. A research engineer is testing the effectiveness of the braking system of a car when it is driven in wet conditions.

The engineer measures and records the braking distance, \(d\) metres, when the brakes are applied from a speed of \(V \mathrm { kmh } ^ { - 1 }\). Graphs of \(d\) against \(V\) and \(\log _ { 10 } d\) against \(\log _ { 10 } V\) were plotted.
The results are shown below together with a data point from each graph. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Explain how Figure 6 would lead the engineer to believe that the braking distance should be modelled by the formula $$d = k V ^ { n } \quad \text { where } k \text { and } n \text { are constants }$$ with \(k \approx 0.017\) Using the information given in Figure 5, with \(k = 0.017\)
2. find a complete equation for the model giving the value of \(n\) to 3 significant figures. Sean is driving this car at \(60 \mathrm { kmh } ^ { - 1 }\) in wet conditions when he notices a large puddle in the road 100 m ahead. It takes him 0.8 seconds to react before applying the brakes.
3. Use your formula to find out if Sean will be able to stop before reaching the puddle.

1. The world human population, \(P\) billions, is modelled by the equation

$$P = a b ^ { t }$$ where \(a\) and \(b\) are constants and \(t\) is the number of years after 2004
Using the estimated population figures for the years from 2004 to 2007, a graph is plotted of \(\log _ { 10 } P\) against \(t\). The points lie approximately on a straight line with

• gradient 0.0054
• intercept 0.81 on the \(\log _ { 10 } P\) axis
  1. Estimate, to 3 decimal places, the value of \(a\) and the value of \(b\).

In the context of the model,

1. interpret the value of the constant \(a\),
2. interpret the value of the constant \(b\).

Use the model to estimate the world human population in 2030

Comment on the reliability of the answer to part (c).

1. The time, \(T\) seconds, that a pendulum takes to complete one swing is modelled by the formula

$$T = a l ^ { b }$$ where \(l\) metres is the length of the pendulum and \(a\) and \(b\) are constants.

1. Show that this relationship can be written in the form $$\log _ { 10 } T = b \log _ { 10 } l + \log _ { 10 } a$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-26_581_888_749_625} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} A student carried out an experiment to find the values of the constants \(a\) and \(b\).
   The student recorded the value of \(T\) for different values of \(l\).
   Figure 3 shows the linear relationship between \(\log _ { 10 } l\) and \(\log _ { 10 } T\) for the student's data.
   The straight line passes through the points \(( - 0.7,0 )\) and \(( 0.21,0.45 )\) Using this information,
2. find a complete equation for the model in the form $$T = a l ^ { b }$$ giving the value of \(a\) and the value of \(b\), each to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(a\).

7 The relationship between the variables \(P\) and \(Q\) is modelled by the formula \(Q = a P ^ { n }\) where \(a\) and \(n\) are constants.
Some values of \(P\) and \(Q\) are obtained from an experiment.

1. State appropriate quantities to plot so that the resulting points lie approximately in a straight line.
2. Explain how to use such a graph to estimate the value of \(n\).

6 During some research the size, \(P\), of a population of insects, at time \(t\) months after the start of the research, is modelled by the following formula. \(P = 100 \mathrm { e } ^ { t }\)

1. Use this model to answer the following.
   1. Find the value of \(P\) when \(t = 4\).
   2. Find the value of \(t\) when the population is 9000 .
2. It is suspected that a more appropriate model would be the following formula. \(P = k a ^ { t }\) where \(k\) and \(a\) are constants.
   1. Show that, using this model, the graph of \(\log _ { 10 } P\) against \(t\) would be a straight line. Some observations of \(t\) and \(P\) gave the following results.

      \(t\)	1	2	3	4	5
      \(P\)	100	500	1800	7000	19000
      \(\log _ { 10 } P\)	2.00	2.70	3.26	3.85	4.28

   2. On the grid in the Printed Answer Booklet, draw a line of best fit for the data points \(\left( t , \log _ { 10 } P \right)\) given in the table.
   3. Hence estimate the values of \(k\) and \(a\).

11 The intensity of the sun's radiation, \(y\) watts per square metre, and the average distance from the sun, \(x\) astronomical units, are shown in Fig. 11 for the planets Mercury and Jupiter. \begin{table}[h] \end{table} The intensity \(y\) is proportional to a power of the distance \(x\).

1. Write down an equation for \(y\) in terms of \(x\) and two constants.
2. Show that the equation can be written in the form \(\ln y = a + b \ln x\).
3. In the Printed Answer Booklet, complete the table for \(\ln x\) and \(\ln y\) correct to 4 significant figures.
4. Use the values from part (iii) to find \(a\) and \(b\).
5. Hence rewrite your equation from part (i) for \(y\) in terms of \(x\), using suitable numerical values for the constants.
6. Sketch a graph of the equation found in part (v).
7. Earth is 1 astronomical unit from the sun. Find the intensity of the sun's radiation for Earth.

11 A student records the time a pendulum takes to swing for different lengths of pendulum. The student decides to plot a graph of \(\log _ { 10 } T\) against \(\log _ { 10 } l\) where \(T\) is the time in seconds that the pendulum takes to return to its start position and \(l\) is the length in metres of the pendulum. They use a model for \(\log _ { 10 } T\) in terms of \(\log _ { 10 } l\) of the form \(\log _ { 10 } T = \log _ { 10 } \mathrm { k } + \mathrm { n } \log _ { 10 } \mathrm { l }\). The student records the following data points.

1. Determine the values of \(k\) and \(n\) that best model the data. Give your values correct to 2 significant figures.
2. Using these values of \(k\) and \(n\), write the student's model as an equation expressing \(T\) in terms of \(l\).