Solve exponential equation by substitution

1 Using the substitution \(u = \mathrm { e } ^ { x }\), or otherwise, solve the equation $$\mathrm { e } ^ { x } = 1 + 6 \mathrm { e } ^ { - x }$$ giving your answer correct to 3 significant figures.

2 Using the substitution \(u = 3 ^ { x }\), solve the equation \(3 ^ { x } + 3 ^ { 2 x } = 3 ^ { 3 x }\) giving your answer correct to 3 significant figures.

2 Using the substitution \(u = 3 ^ { x }\), solve the equation \(3 ^ { x } + 3 ^ { 2 x } = 3 ^ { 3 x }\) giving your answer correct to 3 significant figures.

2

1. Express \(4 ^ { x }\) in terms of \(y\), where \(y = 2 ^ { x }\).
2. Hence find the values of \(x\) that satisfy the equation $$3 \left( 4 ^ { x } \right) - 10 \left( 2 ^ { x } \right) + 3 = 0 ,$$ giving your answers correct to 2 decimal places.

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. By substituting \(p = 3 ^ { x }\), show that the equation $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$ can be rewritten in the form $$9 p ^ { 2 } + 26 p - 3 = 0$$
2. Hence solve $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. By substituting \(p = 2 ^ { x }\), show that the equation $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$ can be written in the form $$4 p ^ { 2 } - 33 p + 8 = 0$$
2. Hence solve $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$

6. (a) Given \(y = 2 ^ { x }\), show that $$2 ^ { 2 x + 1 } - 17 \left( 2 ^ { x } \right) + 8 = 0$$ can be written in the form $$2 y ^ { 2 } - 17 y + 8 = 0$$ (b) Hence solve $$2 ^ { 2 x + 1 } - 17 \left( 2 ^ { x } \right) + 8 = 0$$

1. (a) Sketch the graph of \(y = 7 ^ { x } , x \in \mathbb { R }\), showing the coordinates of any points at which the graph crosses the axes.
   (b) Solve the equation

$$7 ^ { 2 x } - 4 \left( 7 ^ { x } \right) + 3 = 0$$ giving your answers to 2 decimal places where appropriate.

8. (a) Sketch the graph of $$y = 3 ^ { x } , \quad x \in \mathbb { R }$$ showing the coordinates of any points at which the graph crosses the axes.
(b) Use algebra to solve the equation $$3 ^ { 2 x } - 9 \left( 3 ^ { x } \right) + 18 = 0$$ giving your answers to 2 decimal places where appropriate.

6. (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 $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$

1 Solve the equation \(\mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } = 0\).

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

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)

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)

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 $$3 ^ { x + 1 } - 3 ^ { 2 x - 1 } = 6$$