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.

1 Use logarithms to solve the equation \(3 ^ { 4 x + 3 } = 5 ^ { 2 x + 7 }\). Give your answer correct to 3 significant figures. [4]

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.

1

1. Use logarithms to solve the equation \(2 ^ { x } = 20 ^ { 5 }\), giving the answer correct to 3 significant figures.
2. Hence determine the number of integers \(n\) satisfying $$20 ^ { - 5 } < 2 ^ { n } < 20 ^ { 5 }$$

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.

1 Solve the equation \(8 ^ { 3 - 6 x } = 4 \times 5 ^ { - 2 x }\). Give your answer correct to 3 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-02_2718_35_141_2012}

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.
   (a) \(y = a ^ { x }\), where \(a\) is a constant such that \(a > 1\).
   (b) \(y = 2 b ^ { x }\), where \(b\) is a constant such that \(0 < b < 1\).
2. 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.

9 Use logarithms to solve the equation \(5 ^ { 3 x } = 100\). Give your answer correct to 3 decimal places. Section B (36 marks)

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.