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

1 Given that \(2 ^ { y } = 9 ^ { 3 x }\), use logarithms to show that \(y = k x\) and find the value of \(k\) correct to 3 significant figures.

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } - 19 x - a$$ where \(a\) is a constant. It is given that \(( x - 3 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.
3. Hence find the exact values of \(y\) that satisfy the equation \(\mathrm { p } \left( \mathrm { e } ^ { y } + \mathrm { e } ^ { - y } \right) = 0\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5

1. Find the quotient when \(x ^ { 4 } - 32 x + 55\) is divided by \(( x - 2 ) ^ { 2 }\) and show that the remainder is 7 .
2. Factorise \(x ^ { 4 } - 32 x + 48\).
3. Hence solve the equation \(\mathrm { e } ^ { - 12 y } - 32 \mathrm { e } ^ { - 3 y } + 48 = 0\), giving your answer in an exact form.

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + a x ^ { 2 } - 3 x - 4$$ where \(a\) is a constant. It is given that ( \(x - 4\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\).
2. Show that the equation \(\mathrm { p } \left( \mathrm { e } ^ { 3 y } \right) = 0\) has only one real root and find its exact value.

1 Use logarithms to solve the equation \(12 ^ { x } = 3 ^ { 2 x + 1 }\). Give your answer correct to 3 significant figures.

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

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + a x + b$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 5 when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\).

1. Find the values of \(a\) and \(b\).
2. Hence find the exact root of the equation \(\mathrm { p } \left( \mathrm { e } ^ { 2 y } \right) = 0\).

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.

1 Given that $$\ln ( 2 x + 1 ) - \ln ( x - 3 ) = 2$$ find \(x\) in terms of e.

4

1. Solve the equation \(| 2 x - 5 | = | x + 6 |\).
2. Hence find the value of \(y\) such that \(\left| 2 ^ { 1 - y } - 5 \right| = \left| 2 ^ { - y } + 6 \right|\). Give your answer correct to 3 significant figures.

2 Given that \(\frac { 2 ^ { 3 x + 2 } + 8 } { 2 ^ { 3 x } - 7 } = 5\), find the value of \(2 ^ { 3 x }\) and hence, using logarithms, find the value of \(x\) correct to 4 significant figures.

2

1. Sketch, on the same diagram, the graphs of \(y = x + 3\) and \(y = | 2 x - 1 |\).
2. Solve the equation \(x + 3 = | 2 x - 1 |\).
3. Find the value of \(y\) such that \(5 ^ { \frac { 1 } { 2 } y } + 3 = \left| 2 \times 5 ^ { \frac { 1 } { 2 } y } - 1 \right|\). Give your answer correct to 3 significant figures.

2 Use logarithms to solve the equation \(14 \mathrm { e } ^ { - 2 x } = 5 ^ { x + 1 }\), giving your answer correct to 3 significant figures. [4]

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + 23 x ^ { 2 } - a x - 8$$ where \(a\) is a constant. It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\) completely.
2. Hence solve the equation \(\mathrm { p } \left( \mathrm { e } ^ { 4 y } \right) = 0\), giving your answer correct to 3 significant figures.

4

1. Sketch, on the same diagram, the graphs of \(y = | 3 x - 5 |\) and \(y = 2 x + 7\).
2. Solve the equation \(| 3 x - 5 | = 2 x + 7\).
3. Hence solve the equation \(\left| 3 ^ { y + 1 } - 5 \right| = 2 \times 3 ^ { y } + 7\), giving your answer correct to 3 significant figures.

4

1. Sketch, on the same diagram, the graphs of \(y = | 3 - x |\) and \(y = 9 - 2 x\).
2. Solve the inequality \(| 3 - x | > 9 - 2 x\).
3. Use logarithms to solve the inequality \(2 ^ { 3 x - 10 } < 500\). Give your answer in the form \(x < a\), where the value of \(a\) is given correct to 3 significant figures.
4. List the integers that satisfy both of the inequalities \(| 3 - x | > 9 - 2 x\) and \(2 ^ { 3 x - 10 } < 500\).

4

1. Solve the equation \(5 ^ { 2 x } + 5 ^ { x } = 12\), giving your answer correct to 3 significant figures.
2. It is given that \(\ln ( y + 5 ) - \ln y = 2 \ln x\). Express \(y\) in terms of \(x\), in a form not involving logarithms.

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

2

1. Use logarithms to solve the equation \(3 ^ { X } = 8\), giving your answer correct to 2 decimal places.
2. It is given that $$\ln z = \ln ( y + 2 ) - 2 \ln y$$ where \(y > 0\). Express \(z\) in terms of \(y\) in a form not involving logarithms.

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

1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. Find the exact value of the gradient of this line.
2. Calculate the \(x\)-coordinate of the point of intersection of this line with the line \(y = 2 x\), giving your answer correct to 2 decimal places.

2 Use logarithms to solve the equation \(4 ^ { x } = 2 \left( 3 ^ { x } \right)\), giving your answer correct to 3 significant figures.

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.

1 Use logarithms to solve the equation \(3 ^ { x } = 2 ^ { x + 2 }\), giving your answer correct to 3 significant figures.

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.

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.