Simple exponential equation solving

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

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

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

1 Use logarithms to solve the equation \(5 ^ { 2 x - 1 } = 2 \left( 3 ^ { x } \right)\), giving your answer correct to 3 significant figures.

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

1 Use logarithms to solve the equation \(4 ^ { 3 x - 1 } = 3 \left( 5 ^ { x } \right)\), giving your answer correct to 3 decimal places.

1 Use logarithms to solve the equation \(5 ^ { 3 - 2 x } = 4 \left( 7 ^ { x } \right)\), giving your answer correct to 3 decimal places.

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

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

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

1 Use logarithms to solve the equation $$5 ^ { x + 3 } = 7 ^ { x - 1 }$$ giving the answer correct to 3 significant figures.

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

1 Use logarithms to solve the equation $$5 ^ { x + 3 } = 7 ^ { x - 1 }$$ giving the answer correct to 3 significant figures.

1 Solve the equation \(2 \left( 3 ^ { 2 x - 1 } \right) = 4 ^ { x + 1 }\), giving your answer correct to 2 decimal places.

9. (a) Sketch the curve with equation $$y = 3 \times 4 ^ { x }$$ showing the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 6 ^ { 1 - x }\) meets the curve with equation \(y = 3 \times 4 ^ { x }\) at the point \(P\).
(b) Show that the \(x\) coordinate of \(P\) is \(\frac { \log _ { 10 } 2 } { \log _ { 10 } 24 }\)

4. Solve the equation $$5 ^ { x } = 17$$ giving your answer to 3 significant figures.

8

1. On a single diagram, sketch the curves with the following equations. In each case state the coordinates of any points of intersection with the axes.
   1. \(y = a ^ { x }\), where \(a\) is a constant such that \(a > 1\).
   2. \(y = 2 b ^ { x }\), where \(b\) is a constant such that \(0 < b < 1\).
   3. The curves in part (i) intersect at the point \(P\). Prove that the \(x\)-coordinate of \(P\) is $$\frac { 1 } { \log _ { 2 } a - \log _ { 2 } b } .$$

3 U se logarithms to solve the equation \(3 ^ { 2 x + 1 } = 5 ^ { 200 }\), giving the value of \(x\) correct to 3 significant figures.

2

1. Write \(\log _ { 2 } 5 + \log _ { 2 } 1.6\) as an integer.
2. Solve the equation \(2 ^ { x } = 3\), giving your answer correct to 4 decimal places.

2 Solve the equation \(3 ^ { x } = 15\), giving your answer correct to 4 decimal places.

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 \(x = \frac { \log _ { 10 } a } { \log _ { 10 } b }\).

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