1.06f Laws of logarithms: addition, subtraction, power rules

A hot drink when first made has a temperature which is \(65°C\) higher than room temperature. The temperature difference, \(d °C\), between the drink and its surroundings decreases by \(1.7\%\) each minute.

1. Show that 3 minutes after the drink is made, \(d = 61.7\) to 3 significant figures. [2]
2. Write down an expression for the value of \(d\) at time \(n\) minutes after the drink is made, where \(n\) is an integer. [1]
3. Show that when \(d < 3\), \(n\) must satisfy the inequality $$n > \frac{\log_{10} 3 - \log_{10} 65}{\log_{10} 0.983}.$$ Hence find the least integer value of \(n\) for which \(d < 3\). [4]
4. The temperature difference at any time \(t\) minutes after the drink is made can also be expressed as \(d = 65 \times 10^{-kt}\), for some constant \(k\). Use the value of \(d\) for 1 minute after the drink is made to calculate the value of \(k\). Hence find the temperature difference 25.3 minutes after the drink is made. [4]

Use logarithms to solve the equation \(3^{x+1} = 5^{2x}\). Give your answer correct to 3 decimal places. [4]

The thickness of a glacier has been measured every five years from 1960 to 2010. The table shows the reduction in thickness from its measurement in 1960. An exponential model may be used for these data, assuming that the relationship between \(h\) and \(t\) is of the form \(h = a \times 10^{bt}\), where \(a\) and \(b\) are constants to be determined.

1. Show that this relationship may be expressed in the form \(\log_{10} h = mt + c\), stating the values of \(m\) and \(c\) in terms of \(a\) and \(b\). [2]
2. Complete the table of values in the answer book, giving your answers correct to 2 decimal places, and plot the graph of \(\log_{10} h\) against \(t\), drawing by eye a line of best fit. [4]
3. Use your graph to find \(h\) in terms of \(t\) for this model. [4]
4. Calculate by how much the glacier will reduce in thickness between 2010 and 2020, according to the model. [2]
5. Give one reason why this model will not be suitable in the long term. [1]

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]

It is given that \(y = 5^{x-1}\).

1. Show that \(x = 1 + \frac{\ln y}{\ln 5}\). [2]
2. Find an expression for \(\frac{dx}{dy}\) in terms of \(y\). [2]
3. Hence find the exact value of the gradient of the curve \(y = 5^{x-1}\) at the point \((3, 25)\). [2]

Make \(x\) the subject of \(t = \ln \sqrt{\frac{5}{(x-3)}}\). [4]

Solve each equation, giving your answers in exact form.

1. \(e^{4x-3} = 2\) [3]
2. \(\ln (2y - 1) = 1 + \ln (3 - y)\) [4]

Solve each equation, giving your answers in exact form.

1. \(\mathrm{e}^{4x-3} = 2\) [2]
2. \(\ln(2y - 1) = 1 + \ln(3 - y)\) [4]

The variables \(y\) and \(x\) are related by an equation of the form $$y = a(b^x)$$ where \(a\) and \(b\) are positive constants. Let \(Y = \log_{10} y\).

1. Show that there is a linear relationship between \(Y\) and \(x\). [2 marks]
2. The graph of \(Y\) against \(x\), shown below, passes through the points \((0, 2.5)\) and \((5, 0.5)\). \includegraphics{figure_3}
   1. Find the gradient of the line. [1 mark]
   2. Find the value of \(a\) and the value of \(b\), giving each answer to three significant figures. [4 marks]

The differential equation $$\frac{dy}{dx} + \frac{1}{1 - x^2} y = (1 - x)^{\frac{1}{2}}, \quad \text{where } |x| < 1,$$ can be solved by the integrating factor method.

1. Use an appropriate result given in the List of Formulae (MF1) to show that the integrating factor can be written as \(\left(\frac{1 + x}{1 - x}\right)^{\frac{1}{2}}\). [2]
2. Hence find the solution of the differential equation for which \(y = 2\) when \(x = 0\), giving your answer in the form \(y = f(x)\). [6]

Using logarithms, solve the equation $$4^{2x+1} = 5^x,$$ giving your answer correct to 3 significant figures. [3]

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]

Express as a single logarithm $$\log_{10} 2 - \log_{10} x$$ Circle your answer. [1 mark] \(\log_{10} (2 + x)\) \quad \(\log_{10} (2 - x)\) \quad \(\log_{10} (2x)\) \quad \(\log_{10} \left(\frac{2}{x}\right)\)

It is given that $$\ln x - \ln y = 3$$

1. Express \(x\) in terms of \(y\) in a form not involving logarithms. [3 marks]
2. Given also that $$x + y = 10$$ find the exact value of \(y\) and the exact value of \(x\) [3 marks]