log(y) vs x: convert and interpret

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 \(t = 5\)

3. \begin{figure}[h] \end{figure} The line \(l\) in Figure 1 shows a linear relationship between \(\log _ { 10 } y\) and \(x\).
The line passes through the points \(( 0,1.5 )\) and \(( - 4.8,0 )\) as shown.

1. Write down an equation for \(l\).
2. Hence, or otherwise, express \(y\) in the form \(k b ^ { x }\), giving the values of the constants \(k\) and \(b\) to 3 significant figures.

1. The amount of money raised for a charity is being monitored.

The total amount raised in the \(t\) months after monitoring began, \(\pounds D\), is modelled by the equation $$\log _ { 10 } D = 1.04 + 0.38 t$$

1. Write this equation in the form $$D = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give each value to 4 significant figures. When \(t = T\), the total amount of money raised is \(\pounds 45000\) According to the model,
2. find the value of \(T\), giving your answer to 3 significant figures. The charity aims to raise a total of \(\pounds 350000\) within the first 12 months of monitoring.
   According to the model,
3. determine whether or not the charity will achieve its aim.

4. \begin{figure}[h] \end{figure} The number of subscribers to an online video streaming service, \(N\), is modelled by the equation $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants and \(t\) is the number of years since monitoring began.
The line in Figure 1 shows the linear relationship between \(t\) and \(\log _ { 10 } N\) The line passes through the points \(( 0,3.08 )\) and \(( 5,3.85 )\) Using this information,

1. find an equation for this line.
2. Find the value of \(a\) and the value of \(b\), giving your answers to 3 significant figures. When \(t = T\) the number of subscribers is 500000 According to the model,
3. find the value of \(T\)

1. A scientist monitored the growth of bacteria on a dish over a 30 -day period.

The area, \(N \mathrm {~mm} ^ { 2 }\), of the dish covered by bacteria, \(t\) days after monitoring began, is modelled by the equation $$\log _ { 10 } N = 0.0646 t + 1.478 \quad 0 \leqslant t \leqslant 30$$

1. Show that this equation may be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(a\) to the nearest integer and give the value of \(b\) to 3 significant figures.
2. Use the model to find the area of the dish covered by bacteria 30 days after monitoring began. Give your answer, in \(\mathrm { mm } ^ { 2 }\), to 2 significant figures.

1. An area of sea floor is being monitored.

The area of the sea floor, \(S \mathrm {~km} ^ { 2 }\), covered by coral reefs is modelled by the equation $$S = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of years after monitoring began.
Given that $$\log _ { 10 } S = 4.5 - 0.006 t$$

1. find, according to the model, the area of sea floor covered by coral reefs when \(t = 2\)
2. find a complete equation for the model in the form $$S = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(q\)

9 \begin{figure}[h] \end{figure} The graph of \(\log _ { 10 } y\) against \(x\) is a straight line as shown in Fig. 9 .

1. Find the equation for \(\log _ { 10 } y\) in terms of \(x\).
2. Find the equation for \(y\) in terms of \(x\). Section B (36 marks)

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)

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.

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. 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.

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. 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).

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 \(a\).

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 \(t\).

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 \(b\).

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.

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}

1. The number of bacteria on a surface is being monitored.

The number of bacteria, \(N\), on the surface, \(t\) hours after monitoring began is modelled by the equation $$\log _ { 10 } N = 0.35 t + 2$$ Use the equation of the model to answer parts (a) to (c).

1. Find the initial number of bacteria on the surface.
2. Show that the equation of the model can be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(b\) to 2 decimal places.
3. Hence find the rate of growth of bacteria on the surface exactly 5 hours after monitoring began.

6 The table below gives the population of breeding pairs of red kites in Yorkshire from 2001 to 2008. Source: \href{http://www.gigrin.co.uk}{www.gigrin.co.uk}
The following model for the population has been proposed: $$N = a \times b ^ { t } ,$$ where \(N\) is the number of breeding pairs \(t\) years after the year 2000, and \(a\) and \(b\) are constants.

1. Show that the model can be transformed to a linear relationship between \(\log _ { 10 } N\) and \(t\).
2. On graph paper, plot \(\log _ { 10 } N\) against \(t\) and draw by eye a line of best fit. Use your line to estimate the values of \(a\) and \(b\) in the equation for \(N\) in terms of \(t\).
3. What values of \(N\) does the model give for the years 2008 and 2020?
4. In which year will the number of breeding pairs first exceed 500 according to the model?
5. Comment on the suitability of the model to predict the population of breeding pairs of red kites in Yorkshire.

Answer part (ii) of this question on the insert provided. Since 1945 the populations of many countries have been growing. The table shows the estimated population of 15- to 59-year-olds in Africa during the period 1955 to 2005. Source: United Nations Such estimates are used to model future population growth and world needs of resources. One model is \(P = a10^{bt}\), where the population is \(P\) millions, \(t\) is the number of years after 1945 and \(a\) and \(b\) are constants.

1. Show that, using this model, the graph of \(\log_{10} P\) against \(t\) is a straight line of gradient \(b\). State the intercept of this line on the vertical axis. [3]
2. On the insert, complete the table, giving values correct to 2 decimal places, and plot the graph of \(\log_{10} P\) against \(t\). Draw, by eye, a line of best fit on your graph. [3]
3. Use your graph to find the equation for \(P\) in terms of \(t\). [4]
4. Use your results to estimate the population of 15- to 59-year-olds in Africa in 2050. Comment, with a reason, on the reliability of this estimate. [3]

The table shows population data for a country. The data may be represented by an exponential model of growth. Using \(t\) as the number of years after 1960, a suitable model is \(p = a \times 10^{kt}\).

1. Derive an equation for \(\log_{10} p\) in terms of \(a\), \(k\) and \(t\). [2]
2. Complete the table and draw the graph of \(\log_{10} p\) against \(t\), drawing a line of best fit by eye. [3]
3. Use your line of best fit to express \(\log_{10} p\) in terms of \(t\) and hence find \(p\) in terms of \(t\). [4]
4. According to the model, what was the population in 1960? [1]
5. According to the model, when will the population reach 200 million? [3]

Answer the whole of this question on the insert provided. A colony of bats is increasing. The population, \(P\), is modelled by \(P = a \times 10^{bt}\), where \(t\) is the time in years after 2000.

1. Show that, according to this model, the graph of \(\log_{10} P\) against \(t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis. [3]
2. The table gives the data for the population from 2001 to 2005.

   Year	2001	2002	2003	2004	2005
   \(t\)	1	2	3	4	5
   \(P\)	7900	8800	10000	11300	12800

   Complete the table of values on the insert, and plot \(\log_{10} P\) against \(t\). Draw a line of best fit for the data. [3]
3. Use your graph to find the equation for \(P\) in terms of \(t\). [4]
4. Predict the population in 2008 according to this model. [2]

There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012–2013 flu epidemic in the UK were as follows.

Week	1	2	3	4	5	6	7	8	9	10
Number of flu viruses	7	10	24	32	40	38	63	96	234	480

These data may be modelled by an equation of the form \(y = a \times 10^{bt}\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.

1. Explain why this model leads to a straight-line graph of \(\log_{10} y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\). [3]
2. Complete the values of \(\log_{10} y\) in the table, draw the graph of \(\log_{10} y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model. [8]

During the decline of the epidemic, an appropriate model was $$y = 921 \times 10^{-0.137w},$$ where \(y\) is the number of flu viruses detected in week \(w\) of the decline.

1. Use this to find the number of viruses detected in week 4 of the decline. [1]

Scientists observed a colony of seabirds over a period of 10 years starting in 2010. They concluded that the number of birds in the colony, its population \(P\), could be modelled by a formula of the form $$P = a(10^{bt})$$ where \(t\) is the time in years after 2010, and \(a\) and \(b\) are constants.

1. Explain what the value of \(a\) represents. [1 mark]
2. Show that \(\log_{10} P = bt + \log_{10} a\) [2 marks]
3. The table below contains some data collected by the scientists.

   Year	2013	2015
   \(t\)	3
   \(P\)	10200	12800
   \(\log_{10} P\)	4.0086

   1. Complete the table, giving the \(\log_{10} P\) value to 5 significant figures. [1 mark]
   2. Use the data to calculate the value of \(a\) and the value of \(b\). [4 marks]
   3. Use the model to estimate the population of the colony in 2024. [2 marks]
4. 1. State an assumption that must be made in using the model to estimate the population of the colony in 2024. [1 mark]
   2. Hence comment, with a reason, on the reliability of your estimate made in part (c)(iii). [1 mark]