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

Solve the equation \(2^{4x-1} = 3^{5-2x}\), giving your answer in the form \(x = \frac{\log_{10} a}{\log_{10} b}\). [6]

\includegraphics{figure_6} In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 6 shows a sketch of part of the curve with equation $$y = 3x \cdot 2^{2x}.$$ The point \(P(a, 96\sqrt{2})\) lies on the curve.

1. Find the exact value of \(a\). [3]

The curve with equation \(y = 3x \cdot 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).

1. Show that the \(x\) coordinate of \(Q\) is \(\frac{3 + 2\log_2 3}{3 + \log_2 3}\). [4]

The resting metabolic rate, \(R\) ml of oxygen consumed per hour, of a particular species of mammal is modelled by the formula, $$R = aM^b$$ where • \(M\) grams is the mass of the mammal • \(a\) and \(b\) are constants

1. Show that this relationship can be written in the form $$\log_{10} R = b \log_{10} M + \log_{10} a$$ [2] \includegraphics{figure_3} A student gathers data for \(R\) and \(M\) and plots a graph of \(\log_{10} R\) against \(\log_{10} M\) The graph is a straight line passing through points \((0.7, 1.2)\) and \((1.8, 1.9)\) as shown in Figure 3.
2. Using this information, find a complete equation for the model. Write your answer in the form $$R = aM^b$$ giving the value of each of \(a\) and \(b\) to 3 significant figures. [3]
3. With reference to the model, interpret the value of the constant \(a\) [1]

Given that \(p\) is a positive constant,

1. show that $$\sum_{n=1}^{11} \ln(p^n) = k \ln p$$ where \(k\) is a constant to be found, [2]
2. show that $$\sum_{n=1}^{11} \ln(8p^n) = 33\ln(2p^2)$$ [2]
3. Hence find the set of values of \(p\) for which $$\sum_{n=1}^{11} \ln(8p^n) < 0$$ giving your answer in set notation. [2]

Given that the equation $$2\log_2 x = \log_2(kx - 1) + 3,$$ only has one solution, find the value of \(x\). [6]

In part (ii) of this question you must show detailed reasoning.

1. Use logarithms to solve the equation \(8^{2x+1} = 24\), giving your answer to 3 decimal places. [2]
2. Find the values of \(y\) such that $$\log_2(11y - 3) - \log_2 3 - 2\log_2 y = 1, \quad y > \frac{3}{11}$$ [6]

The first term of a geometric progression is \(10\) and the common ratio is \(0.8\).

1. Find the fourth term. [2]
2. Find the sum of the first \(20\) terms, giving your answer correct to \(3\) significant figures. [2]
3. The sum of the first \(N\) terms is denoted by \(S_N\), and the sum to infinity is denoted by \(S_\infty\). Show that the inequality \(S_\infty - S_N < 0.01\) can be written as $$0.8^N < 0.0002,$$ and use logarithms to find the smallest possible value of \(N\). [6]

In this question you must show detailed reasoning. The polynomial \(f(x)\) is given by $$f(x) = x^3 + 6x^2 + x - 4.$$

1. 1. Show that \((x + 1)\) is a factor of \(f(x)\). [1]
   2. Hence find the exact roots of the equation \(f(x) = 0\). [4]
2. 1. Show that the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ can be written in the form \(f(x) = 0\). [3]
   2. Explain why the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ has only one real root and state the exact value of this root. [1]

In this question you must show detailed reasoning. The polynomial f(x) is given by $$f(x) = x^3 + 6x^2 + x - 4.$$

1. 1. Show that \((x + 1)\) is a factor of f(x). [1]
   2. Hence find the exact roots of the equation f(x) = 0. [4]
2. 1. Show that the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ can be written in the form f(x) = 0. [3]
   2. Explain why the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ has only one real root and state the exact value of this root. [1]

A student was asked to solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\). The student's attempt is written out below. \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\) \(4\log_3 x - 3 \log_3 x - 2 = 0\) \(\log_3 x - 2 = 0\) \(\log_3 x = 2\) \(x = 8\)

1. Identify the two mistakes that the student has made. [2]
2. Solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\), giving your answers in an exact form. [4]

In this question you must show detailed reasoning. Solve the following equation for \(x\) in the interval \(0° < x < 180°\) $$1 + \log_3\left(1 + \tan^2 2x\right) = 2\log_3(-4\sin 2x)$$ [8]

In this question you must show detailed reasoning. Solutions using calculator technology are not acceptable. Solve the following equations.

1. \(2\log_3(x + 1) = 1 + \log_3(x + 7)\) [4]
2. \(\log_y\left(\frac{1}{x}\right) = -\frac{3}{2}\) [3]

A student dissolves 0.5 kg of salt in a bucket of water. Water leaks out of a hole in the bucket so the student lets fresh water flow in so that the bucket stays full. They assume that the salty water remaining in the bucket mixes with the fresh water that flows in, so the concentration of salt is uniform throughout the bucket. They model the mass \(M\) kg of salt remaining after \(t\) minutes by \(M = ak^t\) where \(a\) and \(k\) are constants.

1. Show that the model for \(M\) can be rewritten in the form \(\log_{10} M = t\log_{10} k + \log_{10} a\). [1]

The student measures the concentration of salt in the bucket at certain times to estimate the mass of the salt remaining. The results are shown in the table below.

\(t\) minutes	8	13	21	35	50
\(M\) kg	0.4	0.3	0.2	0.1	0.05

The student uses this data and plots \(y = \log_{10} M\) against \(x = t\) using graph drawing software. The software gives \(y = -0.0214x - 0.2403\) for the equation of the line of best fit.

1. 1. Find the values of \(a\) and \(k\) that follow from the equation of the line. [2]
   2. Interpret the value of \(k\) in context. [1]
2. It is known that when \(t = 0\) the mass of salt in the bucket is 0.5 kg. Comment on the accuracy when the model is used to estimate the initial mass of the salt. [1]
3. Use the model to predict the value of \(t\) at which \(M = 0.01\) kg. [2]
4. Rewrite the model for \(M\) in the form \(M = ae^{-ht}\) where \(h\) is a constant to be determined. [2]

Find the values of \(x\) such that $$2\log_3 x - \log_3(x - 2) = 2$$ [5]

Given that \(\log_2 x = a\), find, in terms of \(a\), the simplest form of

1. \(\log_2 (16x)\), [2]
2. \(\log_2 \left(\frac{x^4}{2}\right)\) [3]
3. Hence, or otherwise, solve $$\log_2 (16x) - \log_2 \left(\frac{x^4}{2}\right) = \frac{1}{2},$$ giving your answer in its simplest surd form. [4]

(Total 9 marks)

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. 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]

Two thin poles, \(OA\) and \(BC\), are fixed vertically on horizontal ground. A chain is fixed at \(A\) and \(C\) such that it touches the ground at point \(D\) as shown in the diagram. On a coordinate system the coordinates of \(A\), \(B\) and \(D\) are \((0, 3)\), \((5, 0)\) and \((2, 0)\). \includegraphics{figure_5} It is required to find the height of pole \(BC\) by modelling the shape of the curve that the chain forms. Jofra models the curve using the equation \(y = k \cosh(ax - b) - 1\) where \(k\), \(a\) and \(b\) are positive constants.

1. Determine the value of \(k\). [2]
2. Find the exact value of \(a\) and the exact value of \(b\), giving your answers in logarithmic form. [5]

Holly models the curve using the equation \(y = \frac{1}{4}x^2 - 3x + 3\).

1. Write down the coordinates of the point, \((u, v)\) where \(u\) and \(v\) are both non-zero, at which the two models will agree. [1]
2. Show that Jofra's model and Holly's model disagree in their predictions of the height of pole \(BC\) by \(3.32\)m to 3 significant figures. [3]

The first term of a geometric progression is 12 and the second term is 9.

1. Find the fifth term. [3]

The sum of the first \(N\) terms is denoted by \(S_N\) and the sum to infinity is denoted by \(S_\infty\). It is given that the difference between \(S_\infty\) and \(S_N\) is at most 0.0096.

1. Show that \(\left(\frac{3}{4}\right)^N \leqslant 0.0002\). [3]
2. Use logarithms to find the smallest possible value of \(N\). [2]

Use logarithms to solve the equation \(2^{3x-1} = 3^{x+4}\), giving your answer correct to 3 significant figures. [3]

In this question you must show detailed reasoning. Use logarithms to solve the equation \(3^{2x+1} = 4^{100}\), giving your answer correct to 3 significant figures. [4]

A doctors' surgery starts a campaign to reduce missed appointments. The number of missed appointments for each of the first five weeks after the start of the campaign is shown below.

Number of weeks after the start (\(x\))	1	2	3	4	5
Number of missed appointments (\(y\))	235	149	99	59	38

This data could be modelled by an equation of the form \(y = pq^x\) where \(p\) and \(q\) are constants.

1. Show that this relationship may be expressed in the form \(\log_{10} y = mx + c\), expressing \(m\) and \(c\) in terms of \(p\) and/or \(q\). [2]

The diagram below shows \(\log_{10} y\) plotted against \(x\), for the given data. \includegraphics{figure_5}

1. Estimate the values of \(p\) and \(q\). [3]
2. Use the model to predict when the number of missed appointments will fall below 20. Explain why this answer may not be reliable. [2]