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

4 A painting was valued on 1 April 2001 at \(\pounds 5000\).
The value of this painting is modelled by $$V = A p ^ { t }$$ where \(\pounds V\) is the value \(t\) years after 1 April 2001, and \(A\) and \(p\) are constants.

1. Write down the value of \(A\).
2. According to the model, the value of this painting on 1 April 2011 was \(\pounds 25000\). Using this model:
   1. show that \(p ^ { 10 } = 5\);
   2. use logarithms to find the year in which the painting will be valued at \(\pounds 75000\).
3. A painting by another artist was valued at \(\pounds 2500\) on 1 April 1991. The value of this painting is modelled by $$W = 2500 q ^ { t }$$ where \(\pounds W\) is the value \(t\) years after 1 April 1991, and \(q\) is a constant.
   1. Show that, according to the two models, the value of the two paintings will be the same \(T\) years after 1 April 1991, $$\text { where } T = \frac { \ln \left( \frac { 5 } { 2 } \right) } { \ln \left( \frac { p } { q } \right) }$$
   2. Given that \(p = 1.029 q\), find the year in which the two paintings will have the same value.
      [0pt] [1 mark]

4 In this question you must show detailed reasoning. The cubic polynomial \(6 x ^ { 3 } + k x ^ { 2 } + 57 x - 20\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).

1. Use the factor theorem to show that \(k = - 37\).
2. Using this value of \(k\), factorise \(\mathrm { f } ( x )\) completely.
3. 1. Hence find the three values of \(t\) satisfying the equation \(6 \mathrm { e } ^ { - 3 t } - 37 \mathrm { e } ^ { - 2 t } + 57 \mathrm { e } ^ { - t } - 20 = 0\).
   2. Express the sum of the three values found in part (c)(i) as a single logarithm.

4 In this question you must show detailed reasoning. Solve the simultaneous equations \(\mathrm { e } ^ { x } - 2 \mathrm { e } ^ { y } = 3\) \(\mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { 2 y } = 33\). Give your answer in an exact form.

7. A particular species of orchid is being studied. The population \(p\) at time \(t\) years after the study started is assumed to be $$p = \frac { 2800 a \mathrm { e } ^ { 0.2 t } } { 1 + a \mathrm { e } ^ { 0.2 t } } , \text { where } a \text { is a constant. }$$ Given that there were 300 orchids when the study started,

1. show that \(a = 0.12\),
2. use the equation with \(a = 0.12\) to predict the number of years before the population of orchids reaches 1850 .
3. Show that \(p = \frac { 336 } { 0.12 + \mathrm { e } ^ { - 0.2 t } }\).
4. Hence show that the population cannot exceed 2800.

8. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x + \ln 2 , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto \mathrm { e } ^ { 2 x } , & x \in \mathbb { R } . \end{array}$$

1. Prove that the composite function gf is $$\operatorname { gf } : x \mapsto 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
2. Sketch the curve with equation \(y = \operatorname { gf } ( x )\), and show the coordinates of the point where the curve cuts the \(y\)-axis.
3. Write down the range of gf .
4. Find the value of \(x\) for which \(\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3\), giving your answer to 3 significant figures.

9. (i) Find the exact solutions to the equations

1. \(\ln ( 3 x - 7 ) = 5\),
2. \(3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15\).
   (ii) The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R } , \\ \mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , \quad x > 1 . \end{array}$$
   1. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
   2. Find fg and state its range.

9

1. Solve the equation \(3 \log _ { a } x = \log _ { a } 8\).
2. Show that $$3 \log _ { a } 6 - \log _ { a } 8 = \log _ { a } 27$$
3. 1. The point \(P ( 3 , p )\) lies on the curve \(y = 3 \log _ { 10 } x - \log _ { 10 } 8\). Show that \(p = \log _ { 10 } \left( \frac { 27 } { 8 } \right)\).
   2. The point \(Q ( 6 , q )\) also lies on the curve \(y = 3 \log _ { 10 } x - \log _ { 10 } 8\). Show that the gradient of the line \(P Q\) is \(\log _ { 10 } 2\).

6 The diagram shows a sketch of the curve with equation \(y = 3 \left( 2 ^ { x } + 1 \right)\). \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-5_465_851_390_607} The curve \(y = 3 \left( 2 ^ { x } + 1 \right)\) intersects the \(y\)-axis at the point \(A\).

1. Find the \(y\)-coordinate of the point \(A\).
2. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 6 } 3 \left( 2 ^ { x } + 1 \right) d x\).
3. The line \(y = 21\) intersects the curve \(y = 3 \left( 2 ^ { x } + 1 \right)\) at the point \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$2 ^ { x } = 6$$
   2. Use logarithms to find the \(x\)-coordinate of \(P\), giving your answer to three significant figures.

7. (a) Given that \(y = 3 ^ { x }\), find expressions in terms of \(y\) for

1. \(3 ^ { x + 1 }\),
2. \(3 ^ { 2 x - 1 }\).
   (b) Hence, or otherwise, solve the equation $$3 ^ { x + 1 } - 3 ^ { 2 x - 1 } = 6$$

7

1. Given that \(3 \mathrm { e } ^ { x } = 4\), find the exact value of \(x\).
2. 1. By substituting \(y = \mathrm { e } ^ { x }\), show that the equation \(3 \mathrm { e } ^ { x } + 20 \mathrm { e } ^ { - x } = 19\) can be written as \(3 y ^ { 2 } - 19 y + 20 = 0\).
   2. Hence solve the equation \(3 \mathrm { e } ^ { x } + 20 \mathrm { e } ^ { - x } = 19\), giving your answers as exact values. (3 marks)

4 The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 5 - \mathrm { e } ^ { 3 x } , & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { 2 x - 3 } , & \text { for } x \neq 1.5 \end{array}$$

1. Find the range of f.
2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 0\).
3. Find an expression for \(\operatorname { gg } ( x )\), giving your answer in the form \(\frac { a x + b } { c x + d }\), where \(a , b , c\) and \(d\) are integers.
   [0pt] [3 marks]

4 On 1 January 1900, a sculpture was valued at \(\pounds 80\).
When the sculpture was sold on 1 January 1956, its value was \(\pounds 5000\).
The value, \(\pounds V\), of the sculpture is modelled by the formula \(V = A k ^ { t }\), where \(t\) is the time in years since 1 January 1900 and \(A\) and \(k\) are constants.

1. Write down the value of \(A\).
2. Show that \(k \approx 1.07664\).
3. Use this model to:
   1. show that the value of the sculpture on 1 January 2006 will be greater than £200 000;
   2. find the year in which the value of the sculpture will first exceed \(\pounds 800000\).

4 David is researching changes in the selling price of houses. One particular house was sold on 1 January 1885 for \(\pounds 20\). Sixty years later, on 1 January 1945, it was sold for \(\pounds 2000\). David proposes a model $$P = A k ^ { t }$$ for the selling price, \(\pounds P\), of this house, where \(t\) is the time in years after 1 January 1885 and \(A\) and \(k\) are constants.

1. 1. Write down the value of \(A\).
   2. Show that, to six decimal places, \(k = 1.079775\).
   3. Use the model, with this value of \(k\), to estimate the selling price of this house on 1 January 2008. Give your answer to the nearest \(\pounds 1000\).
2. For another house, which was sold for \(\pounds 15\) on 1 January 1885, David proposes the model $$Q = 15 \times 1.082709 ^ { t }$$ for the selling price, \(\pounds Q\), of this house \(t\) years after 1 January 1885. Calculate the year in which, according to these models, these two houses would have had the same selling price.

8

1. A cup of coffee is cooling down in a room. At time \(t\) minutes after the coffee is made, its temperature is \(x ^ { \circ } \mathrm { C }\), where $$x = 15 + 70 \mathrm { e } ^ { - \frac { t } { 40 } }$$
   1. Find the temperature of the coffee when it is made.
   2. Find the temperature of the coffee 30 minutes after it is made.
   3. Find how long it will take for the coffee to cool down to \(60 ^ { \circ } \mathrm { C }\).
2. 1. Use integration to solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 1 } { 40 } ( x - 15 ) , \quad x > 15$$ given that \(x = 85\) when \(t = 0\), expressing \(t\) in terms of \(x\).
   2. Hence show that \(x = 15 + 70 \mathrm { e } ^ { - \frac { t } { 40 } }\).

4 A biologist is researching the growth of a certain species of hamster. She proposes that the length, \(x \mathrm {~cm}\), of a hamster \(t\) days after its birth is given by $$x = 15 - 12 \mathrm { e } ^ { - \frac { t } { 14 } }$$

1. Use this model to find:
   1. the length of a hamster when it is born;
   2. the length of a hamster after 14 days, giving your answer to three significant figures.
2. 1. Show that the time for a hamster to grow to 10 cm in length is given by \(t = 14 \ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.
   2. Find this time to the nearest day.
3. 1. Show that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 14 } ( 15 - x )$$
   2. Find the rate of growth of the hamster, in cm per day, when its length is 8 cm .
      (1 mark)

8

1. The number of fish in a lake is decreasing. After \(t\) years, there are \(x\) fish in the lake. The rate of decrease of the number of fish is proportional to the number of fish currently in the lake.
   1. Formulate a differential equation, in the variables \(x\) and \(t\) and a constant of proportionality \(k\), where \(k > 0\), to model the rate at which the number of fish in the lake is decreasing.
   2. At a certain time, there were 20000 fish in the lake and the rate of decrease was 500 fish per year. Find the value of \(k\).
2. The equation $$P = 2000 - A \mathrm { e } ^ { - 0.05 t }$$ is proposed as a model for the number of fish, \(P\), in another lake, where \(t\) is the time in years and \(A\) is a positive constant. On 1 January 2008, a biologist estimated that there were 700 fish in this lake.
   1. Taking 1 January 2008 as \(t = 0\), find the value of \(A\).
   2. Hence find the year during which, according to this model, the number of fish in this lake will first exceed 1900.

4 A car depreciates in value according to the model $$V = A k ^ { t }$$ where \(\pounds V\) is the value of the car \(t\) months from when it was new, and \(A\) and \(k\) are constants. Its value when new was \(\pounds 12499\) and 36 months later its value was \(\pounds 7000\).

1. 1. Write down the value of \(A\).
   2. Show that the value of \(k\) is 0.984025 , correct to six decimal places.
2. The value of this car first dropped below \(\pounds 5000\) during the \(n\)th month from new. Find the value of \(n\).

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

1 In this question you must show detailed reasoning. Solve the following equations.

1. \(\frac { x } { x + 1 } - \frac { x - 1 } { x + 2 } = 0\)
2. \(\frac { 8 } { x ^ { 6 } } - \frac { 7 } { x ^ { 3 } } - 1 = 0\)
3. \(3 ^ { x ^ { 2 } - 7 } = \frac { 1 } { 243 }\)

10 As part of an experiment, Zena puts a bucket of hot water outside on a day when the outside temperature is \(0 ^ { \circ } \mathrm { C }\). She measures the temperature of the water after 10 minutes and after 20 minutes. Her results are shown below. Zena models the relationship between \(\theta\), the temperature of the water in \({ } ^ { \circ } \mathrm { C }\), and \(t\), the time in minutes, by $$\theta = A \times 10 ^ { - k t }$$ where \(A\) and \(k\) are constants. 10

1. Using \(t = 0\), explain how the value of \(A\) relates to the experiment. 10
2. Show that $$\log _ { 10 } \theta = \log _ { 10 } A - k t$$ 10
3. Using Zena's results, calculate the values of \(A\) and \(k\).
   10
4. Zena states that the temperature of the water will be less than \(1 ^ { \circ } \mathrm { C }\) after 45 minutes. Determine whether the model supports this statement.
   10
5. Explain why Zena's model is unlikely to accurately give the value of \(\theta\) after 45 minutes.

6 Find the solution to $$5 ^ { ( 2 x + 4 ) } = 9$$ giving your answer in the form \(a + \log _ { 5 } b\), where \(a\) and \(b\) are integers.

6 Show that the equation $$\begin{aligned} & \qquad 2 \log _ { 10 } x = \log _ { 10 } 4 + \log _ { 10 } ( x + 8 ) \\ & \text { has exactly one solution. } \\ & \text { Fully justify your answer. } \end{aligned}$$

6 Show that the solution of the equation $$5 ^ { x } = 3 ^ { x + 4 }$$ can be written as $$x = \frac { \ln 81 } { \ln 5 - \ln 3 }$$ Fully justify your answer.

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.

4 Solve the equation \(2 ^ { 5 x } = 15\), giving the value of \(x\) correct to 3 significant figures.