1.06g Equations with exponentials: solve a^x = b

9 The graph shows the function \(\mathrm { y } = \mathrm { e } ^ { 2 \mathrm { x } }\). \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-6_595_732_322_242}

1. Describe the transformation of the graph of \(y = e ^ { x }\) that gives the graph of \(y = e ^ { 2 x }\). A second function is defined by \(\mathrm { y } = \mathrm { k } + \mathrm { e } ^ { \mathrm { x } }\).
2. A copy of the graph of \(\mathrm { y } = \mathrm { e } ^ { 2 \mathrm { x } }\) is given in the Printed Answer Booklet. Add a sketch of the graph of \(\mathrm { y } = \mathrm { k } + \mathrm { e } ^ { \mathrm { x } }\) in a case where \(k\) is a positive constant.
3. Show that the two graphs do not intersect for values of \(k\) less than \(- \frac { 1 } { 4 }\).
4. In the case where \(k = 2\), show that the only point of intersection occurs when \(x = \ln 2\).

10 Layla invests money in the bank and receives compound interest. The amount \(\pounds L\) that she has after \(t\) years is given by the equation \(\mathrm { L } = 2800 \times 1.023 ^ { \mathrm { t } }\).

1. 1. State the amount she invests.
   2. State the annual rate of interest. Amit invests \(\pounds 3000\) and receives \(2 \%\) compound interest per year. The amount \(\pounds A\) that he has after \(t\) years is given by the equation \(\mathrm { A } = \mathrm { ab } ^ { \mathrm { t } }\).
2. Determine the values of the constants \(a\) and \(b\).
3. Layla and Amit invest their money in the bank at the same time. Determine the value of \(t\) for which Layla and Amit have equal amounts in the bank. Give your answer correct to \(\mathbf { 1 }\) decimal place.

11 In this question you must show detailed reasoning.

1. A student is asked to solve the inequality \(x ^ { \frac { 1 } { 2 } } < 4\). The student argues that \(x ^ { \frac { 1 } { 2 } } < 4 \Leftrightarrow x < 16\), so that the solution is \(\{ x : x < 16 \}\).
   Comment on the validity of the student's argument.
2. Solve the inequality \(\left( \frac { 1 } { 2 } \right) ^ { x } < 4\).
3. Show that the equation \(2 \log _ { 2 } ( x + 8 ) - \log _ { 2 } ( x + 6 ) = 3\) has only one root.

3 You are given that \(y = A e ^ { 0.02 t }\).

• Make \(t\) the subject of the formula.
• Find the value of \(t\) when \(y = 10 ^ { 8 }\) and \(A = 6.62 \times 10 ^ { 7 }\).

5 In 2019 scientists developed a model for comparing the ages of humans and dogs.
According to the model, \(Y = A \ln X + B\) where \(X =\) dog age in years and \(Y =\) human age in years.
For the model, it is known that when \(X = 1 , Y = 31\) and when \(X = 12 , Y = 71\).

1. Find the value of \(B\).
2. Determine the value of \(A\), correct to the nearest whole number. Use the model, with the exact value of \(B\) and the value of \(A\) correct to the nearest whole number, to answer parts (c) and (d).
3. Find the human age corresponding to a dog age of 20 years.
4. Determine the dog age corresponding to a human age of 120 years.

2

1. Express \(2 \log _ { 3 } x + \log _ { 3 } a\) as a single logarithm.
2. Given that \(2 \log _ { 3 } x + \log _ { 3 } a = 2\), express \(x\) in terms of \(a\).

14 Alex places a hot object into iced water and records the temperature \(\theta ^ { \circ } \mathrm { C }\) of the object every minute. The temperature of an object \(t\) minutes after being placed in iced water is modelled by \(\theta = \theta _ { 0 } \mathrm { e } ^ { - k t }\) where \(\theta _ { 0 }\) and \(k\) are constants whose values depend on the characteristics of the object. The temperature of Alex's object is \(82 ^ { \circ } \mathrm { C }\) when it is placed into the water. After 5 minutes the temperature is \(27 ^ { \circ } \mathrm { C }\).

1. Find the values of \(\theta _ { 0 }\) and \(k\) that best model the data.
2. Explain why the model may not be suitable in the long term if Alex does not top up the ice in the water.
3. Show that the model with the values found in part (a) can be written as \(\ln \theta = \mathrm { a } -\) bt where \(a\) and \(b\) are constants to be determined. Ben places a different object into iced water at the same time as Alex. The model for Ben's object is \(\ln \theta = 3.4 - 0.08 t\).
4. Determine each of the following:
   Find this time and the corresponding temperature.

5

1. Make \(y\) the subject of the formula \(\log _ { 10 } ( y - k ) = x \log _ { 10 } 2\), where \(k\) is a positive constant.
2. Sketch the graph of \(y\) against \(x\).

3. (a) Given that \(3 ^ { x } = 9 ^ { y - 1 }\), show that \(x = 2 y - 2\).
(b) Solve the simultaneous equations $$\begin{aligned} & x = 2 y - 2 \\ & x ^ { 2 } = y ^ { 2 } + 7 \end{aligned}$$

7. (a) Given that \(y = 2 ^ { x }\), find expressions in terms of \(y\) for

1. \(2 ^ { x + 2 }\),
2. \(2 ^ { 3 - x }\).
   (b) Show that using the substitution \(y = 2 ^ { x }\), the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$ can be rewritten as $$4 y ^ { 2 } - 33 y + 8 = 0$$ (c) Hence solve the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$

8 The diagram shows a sketch of the curve with equation \(y = 3 ^ { x } + 1\). \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-5_535_1011_411_513} The curve intersects the \(y\)-axis at the point \(A\).

1. Write down the \(y\)-coordinate of point \(A\).
2. 1. Use the trapezium rule with five ordinates (four strips) to find an approximation for \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\), giving your answer to three significant figures.
      (4 marks)
   2. By considering the graph of \(y = 3 ^ { x } + 1\), explain with the aid of a diagram whether your approximation will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\).
      (2 marks)
3. The line \(y = 5\) intersects the curve \(y = 3 ^ { x } + 1\) at the point \(P\). By solving a suitable equation, find the \(x\)-coordinate of the point \(P\). Give your answer to four decimal places.
   (4 marks)
4. The curve \(y = 3 ^ { x } + 1\) is reflected in the \(y\)-axis to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
   (1 mark)

3

1. Use logarithms to solve the equation \(0.8 ^ { x } = 0.05\), giving your answer to three decimal places.
2. An infinite geometric series has common ratio \(r\). The sum to infinity of the series is five times the first term of the series.
   1. Show that \(r = 0.8\).
   2. Given that the first term of the series is 20 , find the least value of \(n\) such that the \(n\)th term of the series is less than 1 .

8

1. Sketch the graph of \(y = 3 ^ { x }\), stating the coordinates of the point where the graph crosses the \(y\)-axis.
2. Describe a single geometrical transformation that maps the graph of \(y = 3 ^ { x }\) :
   1. onto the graph of \(y = 3 ^ { 2 x }\);
   2. onto the graph of \(y = 3 ^ { x + 1 }\).
3. 1. Using the substitution \(Y = 3 ^ { x }\), show that the equation $$9 ^ { x } - 3 ^ { x + 1 } + 2 = 0$$ can be written as $$( Y - 1 ) ( Y - 2 ) = 0$$
   2. Hence show that the equation \(9 ^ { x } - 3 ^ { x + 1 } + 2 = 0\) has a solution \(x = 0\) and, by using logarithms, find the other solution, giving your answer to four decimal places.
      (4 marks)

6

1. Write each of the following in the form \(\log _ { a } k\), where \(k\) is an integer:
   1. \(\log _ { a } 4 + \log _ { a } 10\);
   2. \(\log _ { a } 16 - \log _ { a } 2\);
   3. \(3 \log _ { a } 5\).
2. Use logarithms to solve the equation \(( 1.5 ) ^ { 3 x } = 7.5\), giving your value of \(x\) to three decimal places.
3. Given that \(\log _ { 2 } p = m\) and \(\log _ { 8 } q = n\), express \(p q\) in the form \(2 ^ { y }\), where \(y\) is an expression in \(m\) and \(n\).

7

1. Sketch the graph of \(y = \frac { 1 } { 2 ^ { x } }\), indicating the value of the intercept on the \(y\)-axis.
2. Use logarithms to solve the equation \(\frac { 1 } { 2 ^ { x } } = \frac { 5 } { 4 }\), giving your answer to three significant figures.
3. Given that $$\log _ { a } \left( b ^ { 2 } \right) + 3 \log _ { a } y = 3 + 2 \log _ { a } \left( \frac { y } { a } \right)$$ express \(y\) in terms of \(a\) and \(b\).
   Give your answer in a form not involving logarithms.

7

1. Describe a geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 3 \times 4 ^ { x }\).
2. Sketch the curve with equation \(y = 3 \times 4 ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
3. The curve with equation \(y = 4 ^ { - x }\) intersects the curve \(y = 3 \times 4 ^ { x }\) at the point \(P\). Use logarithms to find the \(x\)-coordinate of \(P\), giving your answer to three significant figures.

6 The diagram shows a sketch of the curve with equation \(y = 27 - 3 ^ { x }\). \includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-4_933_1074_376_484} The curve \(y = 27 - 3 ^ { x }\) intersects the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\).

1. 1. Find the \(y\)-coordinate of point \(A\).
   2. Verify that the \(x\)-coordinate of point \(B\) is 3 .
2. The region, \(R\), bounded by the curve \(y = 27 - 3 ^ { x }\) and the coordinate axes is shaded. Use the trapezium rule with four ordinates (three strips) to find an approximate value for the area of \(R\).
3. 1. Use logarithms to solve the equation \(3 ^ { x } = 13\), giving your answer to four decimal places.
   2. The line \(y = k\) intersects the curve \(y = 27 - 3 ^ { x }\) at the point where \(3 ^ { x } = 13\). Find the value of \(k\).
4. 1. Describe the single geometrical transformation by which the curve with equation \(y = - 3 ^ { x }\) can be obtained from the curve \(y = 27 - 3 ^ { x }\).
   2. Sketch the curve \(y = - 3 ^ { x }\).

8 The diagram shows a sketch of the curve with equation \(y = 6 ^ { x }\). \includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-5_403_506_370_769}

1. 1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
   2. Explain, with the aid of a diagram, whether your approximate value will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\).
2. 1. Describe a single geometrical transformation that maps the graph of \(y = 6 ^ { x }\) onto the graph of \(y = 6 ^ { 3 x }\).
   2. The line \(y = 84\) intersects the curve \(y = 6 ^ { 3 x }\) at the point \(A\). By using logarithms, find the \(x\)-coordinate of \(A\), giving your answer to three decimal places.
      (4 marks)
3. The graph of \(y = 6 ^ { x }\) is translated by \(\left[ \begin{array} { c } 1 \\ - 2 \end{array} \right]\) to give the graph of the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).

8 The diagram shows a sketch of the curve \(y = 2 ^ { 4 x }\). \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-9_435_814_374_623} The curve intersects the \(y\)-axis at the point \(A\).

1. Find the value of the \(y\)-coordinate of \(A\).
2. Use the trapezium rule with six ordinates (five strips) to find an approximate value for \(\int _ { 0 } ^ { 1 } 2 ^ { 4 x } \mathrm {~d} x\), giving your answer to two decimal places.
3. Describe the geometrical transformation that maps the graph of \(y = 2 ^ { 4 x }\) onto the graph of \(y = 2 ^ { 4 x - 3 }\).
4. The curve \(y = 2 ^ { 4 x }\) is translated by the vector \(\left[ \begin{array} { c } 1 \\ - \frac { 1 } { 2 } \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\). The curve \(y = \mathrm { g } ( x )\) crosses the \(x\)-axis at the point \(Q\). Find the \(x\)-coordinate of \(Q\).
5. 1. Given that $$\log _ { a } k = 3 \log _ { a } 2 + \log _ { a } 5 - \log _ { a } 4$$ show that \(k = 10\).
   2. The line \(y = \frac { 5 } { 4 }\) crosses the curve \(y = 2 ^ { 4 x - 3 }\) at the point \(P\). Show that the \(x\)-coordinate of \(P\) is \(\frac { 1 } { 4 \log _ { 10 } 2 }\).

4

1. Sketch the curve with equation \(y = 4 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
   (2 marks)
2. Describe the geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 4 ^ { x } - 5\).
3. 1. Use the substitution \(Y = 2 ^ { x }\) to show that the equation \(4 ^ { x } - 2 ^ { x + 2 } - 5 = 0\) can be written as \(Y ^ { 2 } - 4 Y - 5 = 0\).
   2. Hence show that the equation \(4 ^ { x } - 2 ^ { x + 2 } - 5 = 0\) has only one real solution. Use logarithms to find this solution, giving your answer to three decimal places.
      (4 marks)

8

1. Sketch the curve with equation \(y = 7 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
2. The curve \(C _ { 1 }\) has equation \(y = 7 ^ { x }\). The curve \(C _ { 2 }\) has equation \(y = 7 ^ { 2 x } - 12\).
   1. By forming and solving a quadratic equation, prove that the curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at exactly one point. State the \(y\)-coordinate of this point.
   2. Use logarithms to find the \(x\)-coordinate of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\), giving your answer to three significant figures.
      (2 marks)

4

1. Sketch the graph of \(y = 9 ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
   (2 marks)
2. Use logarithms to solve the equation \(9 ^ { x } = 15\), giving your value of \(x\) to three significant figures.
3. The curve \(y = 9 ^ { x }\) is reflected in the \(y\)-axis to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
   (l mark)

9 A curve has equation \(y = 3 \times 12 ^ { x }\).

1. The point ( \(k , 6\) ) lies on the curve \(y = 3 \times 12 ^ { x }\). Use logarithms to find the value of \(k\), giving your answer to three significant figures.
2. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 1.5 } 3 \times 12 ^ { x } \mathrm {~d} x\), giving your answer to two significant figures.
3. The curve \(y = 3 \times 12 ^ { x }\) is translated by the vector \(\left[ \begin{array} { l } 1 \\ p \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the curve \(y = \mathrm { f } ( x )\) passes through the origin ( 0,0 ), find the value of the constant \(p\).
4. The curve with equation \(y = 2 ^ { 2 - x }\) intersects the curve \(y = 3 \times 12 ^ { x }\) at the point \(T\). Show that the \(x\)-coordinate of \(T\) can be written in the form \(\frac { 2 - \log _ { 2 } 3 } { q + \log _ { 2 } 3 }\), where \(q\) is an integer. State the value of \(q\).
   [0pt] [5 marks]
   \includegraphics[max width=\textwidth, alt={}]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-20_2288_1707_221_153}

9

1. Use logarithms to solve the equation \(2 ^ { 3 x } = 5\), giving your value of \(x\) to three significant figures.
2. Given that \(\log _ { a } k - \log _ { a } 2 = \frac { 2 } { 3 }\), express \(a\) in terms of \(k\).
3. 1. By using the binomial expansion, or otherwise, express \(( 1 + 2 x ) ^ { 3 }\) in ascending powers of \(x\).
   2. It is given that $$\log _ { 2 } \left[ ( 1 + 2 n ) ^ { 3 } - 8 n \right] = \log _ { 2 } ( 1 + 2 n ) + \log _ { 2 } \left[ 4 \left( 1 + n ^ { 2 } \right) \right]$$ By forming and solving a suitable quadratic equation, find the possible values of \(n\). [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-20_1581_1714_1126_153} \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-24_2488_1728_219_141}

2

1. Sketch the graph of \(y = ( 0.2 ) ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
2. Use logarithms to solve the equation \(( 0.2 ) ^ { x } = 4\), giving your answer to three significant figures.
3. Describe the geometrical transformation that maps the graph of \(y = ( 0.2 ) ^ { x }\) onto the graph of \(y = 5 ^ { x }\).
   [0pt] [1 mark]