Exponential Equations & Modelling

Use logarithms to solve the equation \(2^{2x-1} = 5\). [4]

1. Do not use a calculator for this question

Find the value of \(x\) for which \(\sqrt { 3 } \times 3 ^ { x } = \frac { 1 } { 9 }\) [0pt] [2 marks]

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1. On a single diagram, sketch the curves with the following equations. In each case state the coordinates of any points of intersection with the axes.
   1. \(y = a ^ { x }\), where \(a\) is a constant such that \(a > 1\).
   2. \(y = 2 b ^ { x }\), where \(b\) is a constant such that \(0 < b < 1\).
   3. The curves in part (i) intersect at the point \(P\). Prove that the \(x\)-coordinate of \(P\) is $$\frac { 1 } { \log _ { 2 } a - \log _ { 2 } b } .$$

8

1. Solve the equation \(10 ^ { x } = 316\).
2. Simplify \(\log _ { a } \left( a ^ { 2 } \right) - 4 \log _ { a } \left( \frac { 1 } { a } \right)\).

3. Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which

1. \(5 ^ { x } = 10\),
2. \(\log _ { 3 } ( x - 2 ) = - 1\).

1.(a)By writing \(u = \log _ { 4 } r\) ,where \(r > 0\) ,show that $$\log _ { 4 } r = \frac { 1 } { 2 } \log _ { 2 } r$$ (b)Solve the equation $$\log _ { 4 } \left( 5 x ^ { 2 } - 11 \right) = \log _ { 2 } ( 3 x - 5 )$$

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)

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]

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 \(T\)

Solve the equation \(5^x = 5^{x+2} - 10\). Give your answer correct to 3 decimal places. [3]

5

1. Given that \(y = 2 ^ { x }\), show that the equation $$2 ^ { x } + 3 \left( 2 ^ { - x } \right) = 4$$ can be written in the form $$y ^ { 2 } - 4 y + 3 = 0$$
2. Hence solve the equation $$2 ^ { x } + 3 \left( 2 ^ { - x } \right) = 4$$ giving the values of \(x\) correct to 3 significant figures where appropriate.

3

1. Show that the equation $$\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0$$ can be expressed as a quadratic equation in \(\mathrm { e } ^ { x }\).
2. Hence solve the equation \(\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0\), giving your answer correct to 3 decimal places.

\includegraphics{figure_2} The variables \(x\) and \(y\) satisfy the equation \(y = Ae^{px}\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((5, 3.17)\) and \((10, 4.77)\), as shown in the diagram. Find the values of \(A\) and \(p\) correct to 2 decimal places. [5]

3 The variables \(x\) and \(y\) satisfy the equation \(y = 3 ^ { 2 a } a ^ { x }\), where \(a\) is a constant. The graph of \(\ln y\) against \(x\) is a straight line with gradient 0.239 .

1. Find the value of \(a\) correct to 3 significant figures.
2. Hence find the value of \(x\) when \(y = 36\). Give your answer correct to 3 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{de8af872-9f77-4787-8e66-ed199405ca25-2_583_597_1457_772} The variables \(x\) and \(y\) satisfy the equation \(y = K \left( 2 ^ { p x } \right)\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(1.35,1.87\) ) and ( \(3.35,3.81\) ), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places.
[0pt] [6]

3. Find the exact solutions of

1. \(\mathrm { e } ^ { 2 x + 3 } = 6\),
2. \(\ln ( 3 x + 2 ) = 4\).

3. Find the exact solutions of

1. \(\mathrm { e } ^ { 2 x + 3 } = 6\),
2. \(\ln ( 3 x + 2 ) = 4\).

3. Giving your answers to 2 decimal places, solve the simultaneous equations $$\begin{aligned} & \mathrm { e } ^ { 2 y } - x + 2 = 0 \\ & \ln ( x + 3 ) - 2 y - 1 = 0 \end{aligned}$$

The temperature \(\theta\) °C of water in a container after \(t\) minutes is modelled by the equation $$\theta = a - be^{-kt},$$ where \(a\), \(b\) and \(k\) are positive constants. The initial and long-term temperatures of the water are 15°C and 100°C respectively. After 1 minute, the temperature is 30°C.

1. Find \(a\), \(b\) and \(k\). [6]
2. Find how long it takes for the temperature to reach 80°C. [2]

6 An on-line science website states:
'To find a dog's equivalent human age in years, multiply the natural logarithm of the dog's age in years by 16 then add 31.' 6

1. Calculate the equivalent age to the nearest human year of a dog aged 5 years. 6
2. A dog's equivalent age in human years is 40 years. Find the dog's actual age to the nearest month.
   6
3. Explain why the behaviour of the natural logarithm for values close to zero means that the formula given on the website cannot be true for very young dogs.

1. The value of a car is modelled by the formula

$$V = 16000 \mathrm { e } ^ { - k t } + A , \quad t \geqslant 0 , t \in \mathbb { R }$$ where \(V\) is the value of the car in pounds, \(t\) is the age of the car in years, and \(k\) and \(A\) are positive constants. Given that the value of the car is \(\pounds 17500\) when new and \(\pounds 13500\) two years later,

1. find the value of \(A\),
2. show that \(k = \ln \left( \frac { 2 } { \sqrt { 3 } } \right)\)
3. Find the age of the car, in years, when the value of the car is \(\pounds 6000\) Give your answer to 2 decimal places.

3 The variables \(x\) and \(y\) satisfy the equation \(x ^ { n } y = C\), where \(n\) and \(C\) are constants. When \(x = 1.10\), \(y = 5.20\), and when \(x = 3.20 , y = 1.05\).

1. Find the values of \(n\) and \(C\).
2. Explain why the graph of \(\ln y\) against \(\ln x\) is a straight line.

3 Two variable quantities \(x\) and \(y\) are related by the equation $$y = A x ^ { n }$$ where \(A\) and \(n\) are constants. \includegraphics[max width=\textwidth, alt={}, center]{9b103197-7ba0-427a-b983-34edb51b6cca-2_422_697_977_740} When a graph is plotted showing values of \(\ln y\) on the vertical axis and values of \(\ln x\) on the horizontal axis, the points lie on a straight line. This line crosses the vertical axis at the point ( \(0,2.3\) ) and also passes through the point (4.0,1.7), as shown in the diagram. Find the values of \(A\) and \(n\).

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 \(p\).

The temperature \(T°C\) of a liquid at time \(t\) minutes is given by the equation $$T = 30 + 20e^{-0.05t}, \quad \text{for } t \geq 0.$$ Write down the initial temperature of the liquid, and find the initial rate of change of temperature. Find the time at which the temperature is \(40°C\).

The temperature \(T°C\) of a liquid at time \(t\) minutes is given by the equation $$T = 30 + 20e^{-0.05t}, \quad \text{for } t \geq 0.$$ Write down the initial temperature of the liquid, and find the initial rate of change of temperature. Find the time at which the temperature is \(40°C\).

11 In a science experiment a substance is decaying exponentially. Its mass, \(M\) grams, at time \(t\) minutes is given by \(M = 300 e ^ { - 0.05 t }\).

1. Find the time taken for the mass to decrease to half of its original value. A second substance is also decaying exponentially. Initially its mass was 400 grams and, after 10 minutes, its mass was 320 grams.
2. Find the time at which both substances are decaying at the same rate.

1 Given that \(2 ^ { x } = 5 ^ { y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 3 significant figures.

8 Using logarithms, rearrange \(p = s t ^ { n }\) to make \(n\) the subject.

4 The variables \(x\) and \(y\) satisfy the equation \(5 ^ { y + 1 } = 2 ^ { 3 x }\).

1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line.
2. Find the exact value of the gradient of this line and state the coordinates of the point at which the line cuts the \(y\)-axis.

3 \includegraphics[max width=\textwidth, alt={}, center]{83d0697c-b133-47da-a745-dfdafa7dbf10-05_604_933_258_605} The variables \(x\) and \(y\) satisfy the equation \(a ^ { y } = k x\), where \(a\) and \(k\) are constants. The graph of \(y\) against \(\ln x\) is a straight line passing through the points \(( 1.03,6.36 )\) and \(( 2.58,9.00 )\), as shown in the diagram. Find the values of \(a\) and \(k\), giving each value correct to 2 significant figures.

1 Solve the equation $$\frac { 2 ^ { x } + 1 } { 2 ^ { x } - 1 } = 5$$ giving your answer correct to 3 significant figures.

Use logarithms to solve the equation \(6^{2x-1} = 5e^{3x+2}\). Give your answer correct to 4 significant figures. [4]

\includegraphics{figure_3} The variables \(x\) and \(y\) satisfy the equation \(y = Ae^{-kx^2}\), where \(A\) and \(k\) are constants. The graph of \(\ln y\) against \(x^2\) is a straight line passing through the points \((0.64, 0.76)\) and \((1.69, 0.32)\), as shown in the diagram. Find the values of \(A\) and \(k\) correct to 2 decimal places. [5]

1 Find the set of values of \(x\) for which \(2 \left( 3 ^ { 1 - 2 x } \right) < 5 ^ { x }\). Give your answer in a simplified exact form. [4]