Quadratic in exponential form

4

1. Sb the eq tiந \({ } ^ { 2 x } + 5 ^ { x } = \frac { 13 } { 8 } \quad\) in wer co rect t \(\beta \quad\) sig fican fig es. [44
2. It is g vert \(\mathbf { h } \mathrm { t } \ln y + \overline { 5 } + \mathrm { n } y = 2 \ln x\). Eq ess \(y\) irt erms \(\mathbf { b } x\), irr fo m no id \(\mathbf { v }\) ng \(\mathbf { g }\) riths.

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.

2

1. Given that \(5 ^ { 2 x } + 5 ^ { x } = 12\), find the value of \(5 ^ { x }\).
2. Hence, using logarithms, solve the equation \(5 ^ { 2 x } + 5 ^ { x } = 12\), giving the value of \(x\) correct to 3 significant figures.

3 Given that \(3 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x } = 14\), find the possible values of \(\mathrm { e } ^ { x }\) and hence solve the equation \(3 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x } = 14\) correct to 3 significant figures.

1 Solve the equation \(3 \mathrm { e } ^ { 2 x } - 82 \mathrm { e } ^ { x } + 27 = 0\), giving your answers in the form \(k \ln 3\).

2 Using the substitution \(u = 4 ^ { x }\), solve the equation \(4 ^ { x } + 4 ^ { 2 } = 4 ^ { x + 2 }\), giving your answer correct to 3 significant figures.

3 Using the substitution \(u = \mathrm { e } ^ { x }\), solve the equation \(4 \mathrm { e } ^ { - x } = 3 \mathrm { e } ^ { x } + 4\). Give your answer correct to 3 significant figures.

2 Showing all necessary working, solve the equation \(9 ^ { x } = 3 ^ { x } + 12\). Give your answer correct to 2 decimal places.

3

1. Express \(9 ^ { x }\) in terms of \(y\), where \(y = 3 ^ { x }\).
2. Hence solve the equation $$2 \left( 9 ^ { x } \right) - 7 \left( 3 ^ { x } \right) + 3 = 0 ,$$ expressing your answers for \(x\) in terms of logarithms where appropriate.

4 Solve the equation \(3 ^ { 2 x } - 7 \left( 3 ^ { x } \right) + 10 = 0\), giving your answers correct to 3 significant figures.

1

1. It is given that \(x\) satisfies the equation \(3 ^ { 2 x } = 5 \left( 3 ^ { x } \right) + 14\). Find the value of \(3 ^ { x }\) and, using logarithms, find the value of \(x\) correct to 3 significant figures.
2. Hence state the values of \(x\) satisfying the equation \(3 ^ { 2 | x | } = 5 \left( 3 ^ { | x | } \right) + 14\).

2 Given that \(9 ^ { x } + 3 ^ { x } = 240\), find the value of \(3 ^ { x }\) and hence, using logarithms, find the value of \(x\) correct to 4 significant figures.

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.

2 Solve the equation \(4 ^ { x } = 3 + 4 ^ { - x }\). Give your answer correct to 3 decimal places.

1 Solve the equation $$3 \mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { - 2 x } = 5$$ Give the answer correct to 3 decimal places.

4. (a) Find, to 3 significant figures, the value of \(x\) for which \(5 ^ { x } = 7\).
(b) Solve the equation \(5 ^ { 2 x } - 12 \left( 5 ^ { x } \right) + 35 = 0\).

5. (i) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (ii) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0$$

12. A student was asked to give the exact solution to the equation $$2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0$$ The student's attempt is shown below: $$\begin{aligned} & 2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0 \\ & 2 ^ { 2 x } + 2 ^ { 4 } - 9 \left( 2 ^ { x } \right) = 0 \\ & \text { Let } \quad 2 ^ { x } = y \\ & y ^ { 2 } - 9 y + 8 = 0 \\ & ( y - 8 ) ( y - 1 ) = 0 \\ & y = 8 \text { or } y = 1 \\ & \text { So } x = 3 \text { or } x = 0 \end{aligned}$$

1. Identify the two errors made by the student.
2. Find the exact solution to the equation.

8. \begin{figure}[h] \end{figure} The curves with equation \(y = 21 - 2 ^ { x }\) meet the curve with equation \(y = 2 ^ { 2 x + 1 }\) at the point \(A\) as shown in Figure 2. Find the exact coordinates of point \(A\).

5. (a) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (b) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0 .$$

5

1. Solve the equation \(2 \mathrm { e } ^ { x } = 5\), giving your answer as an exact natural logarithm.
2. 1. By substituting \(y = \mathrm { e } ^ { x }\), show that the equation \(2 \mathrm { e } ^ { x } + 5 \mathrm { e } ^ { - x } = 7\) can be written as $$2 y ^ { 2 } - 7 y + 5 = 0$$
   2. Hence solve the equation \(2 \mathrm { e } ^ { x } + 5 \mathrm { e } ^ { - x } = 7\), giving your answers as exact values of \(x\).

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)

8

1. Using \(y = 2 ^ { 2 x }\) as a substitution, show that $$16 ^ { x } - 2 ^ { ( 2 x + 3 ) } - 9 = 0$$ can be written as $$y ^ { 2 } - 8 y - 9 = 0$$ 8
2. Hence, show that the equation $$16 ^ { x } - 2 ^ { ( 2 x + 3 ) } - 9 = 0$$ has \(x = \log _ { 2 } 3\) as its only solution.
   Fully justify your answer.