Simple exponential equation solving

9

1. Sketch the curve \(y = 6 \times 5 ^ { x }\), stating the coordinates of any points of intersection with the axes.
2. The point \(P\) on the curve \(y = 9 ^ { x }\) has \(y\)-coordinate equal to 150 . Use logarithms to find the \(x\)-coordinate of \(P\), correct to 3 significant figures.
3. The curves \(y = 6 \times 5 ^ { x }\) and \(y = 9 ^ { x }\) intersect at the point \(Q\). Show that the \(x\)-coordinate of \(Q\) can be written as \(x = \frac { 1 + \log _ { 3 } 2 } { 2 - \log _ { 3 } 5 }\).

3 Use logarithms to solve the equation \(7 ^ { x } = 2 ^ { x + 1 }\), giving the value of \(x\) correct to 3 significant figures.

5 Solve the equation \(2 ^ { 4 x - 1 } = 3 ^ { 5 - 2 x }\), giving your answer in the form \(x = \frac { \log _ { 10 } a } { \log _ { 10 } b }\).

9

1. Sketch the graph of \(y = 3 ^ { x }\).
2. Use logarithms to solve \(3 ^ { 2 x + 1 } = 10\), giving your answer correct to 2 decimal places.

6 Use logarithms to solve the equation \(235 \times 5 ^ { x } = 987\), giving your answer correct to 3 decimal places.

8

1. Sketch the graph of \(y = 3 ^ { x }\).
2. Solve the equation \(3 ^ { 5 x - 1 } = 500000\).

10 Use logarithms to solve the equation \(3 ^ { x + 1 } = 5 ^ { 2 x }\). Give your answer correct to 3 decimal places. Section B (36 marks) \begin{figure}[h] \end{figure} Fig. 11 shows a sketch of the curve with equation \(y = x - \frac { 4 } { x ^ { 2 } }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 24 } { x ^ { 4 } }\).
2. Hence find the coordinates of the stationary point on the curve. Verify that the stationary point is a maximum.
3. Find the equation of the normal to the curve when \(x = - 1\). Give your answer in the form \(a x + b y + c = 0\).

8

1. Simplify \(\log _ { a } 1 - \log _ { a } \left( a ^ { m } \right) ^ { 3 }\).
2. Use logarithms to solve the equation \(3 ^ { 2 x + 1 } = 1000\). Give your answer correct to 3 significant figures.

1 Using logarithms, solve the equation \(4 ^ { 2 x + 1 } = 5 ^ { x }\),
giving your answer correct to \(\mathbf { 3 }\) significant figures.

2. The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$\begin{aligned} & C _ { 1 } : \quad y = 2 ^ { 3 x + 2 } \\ & C _ { 2 } : \quad y = 4 ^ { - x } \end{aligned}$$ Show that the \(x\)-coordinate of the point where \(C _ { 1 }\) and \(C _ { 2 }\) intersect is \(\frac { - 2 } { 5 }\).

2. Solve $$4 ^ { x - 3 } = 6$$ giving your answer in the form \(a + b \log _ { 2 } 3\), where \(a\) and \(b\) are constants to be found.

1. By taking logarithms of both sides, solve the equation

$$4 ^ { 3 p - 1 } = 5 ^ { 210 }$$ giving the value of \(p\) to one decimal place.

3

1. On the axes in the Printed Answer Booklet, sketch the curve with equation \(\mathrm { y } = 3 \times 0.4 ^ { \mathrm { x } }\).
2. Given that \(3 \times 0.4 ^ { x } = 0.8\), determine the value of \(x\) correct to 3 significant figures.

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 .

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}

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

3

1. Write down the values of \(p , q\) and \(r\) given that:
   1. \(64 = 8 ^ { p }\);
   2. \(\frac { 1 } { 64 } = 8 ^ { q }\);
   3. \(\sqrt { 8 } = 8 ^ { r }\).
2. Find the value of \(x\) for which $$\frac { 8 ^ { x } } { \sqrt { 8 } } = \frac { 1 } { 64 }$$

9 Use logarithms to solve the equation \(5 ^ { x } = 235\), giving your answer correct to 2 decimal places.

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

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 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.

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.