Solve exponential equation using logarithms

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

2 Solve the equation $$5 ^ { x - 1 } = 5 ^ { x } - 5$$ giving your answer correct to 3 significant figures.

1 Find the value of \(x\) for which \(3 \left( 2 ^ { 1 - x } \right) = 7 ^ { x }\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.

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

3 Solve the equation \(2 ^ { 3 x - 1 } = 5 \left( 3 ^ { - x } \right)\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.

1 Solve the equation \(2 ^ { 3 x - 1 } = 5 \left( 3 ^ { 1 - x } \right)\). Give your answer in the form \(\frac { \ln a } { \ln b }\) where \(a\) and \(b\) are integers.

4 Solve the equation \(5 ^ { x } = 5 ^ { x + 2 } - 10\). Give your answer correct to 3 decimal places.

5. Solve

1. \(2 ^ { y } = 8\)
2. \(2 ^ { x } \times 4 ^ { x + 1 } = 8\)

6 Sketch the graph of \(y = 2 ^ { x }\).
Solve the equation \(2 ^ { x } = 50\), giving your answer correct to 2 decimal places.

7

1. Sketch the graph of \(y = 3 ^ { x }\).
2. Use logarithms to solve the equation \(3 ^ { x } = 20\). Give your answer correct to 2 decimal places.

6 The diagram shows a sketch of the curve with equation \(y = 27 - 3 ^ { x }\). \includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-4_933_1074_376_484} The curve \(y = 27 - 3 ^ { x }\) intersects the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\).

1. 1. Find the \(y\)-coordinate of point \(A\).
   2. Verify that the \(x\)-coordinate of point \(B\) is 3 .
2. The region, \(R\), bounded by the curve \(y = 27 - 3 ^ { x }\) and the coordinate axes is shaded. Use the trapezium rule with four ordinates (three strips) to find an approximate value for the area of \(R\).
3. 1. Use logarithms to solve the equation \(3 ^ { x } = 13\), giving your answer to four decimal places.
   2. The line \(y = k\) intersects the curve \(y = 27 - 3 ^ { x }\) at the point where \(3 ^ { x } = 13\). Find the value of \(k\).
4. 1. Describe the single geometrical transformation by which the curve with equation \(y = - 3 ^ { x }\) can be obtained from the curve \(y = 27 - 3 ^ { x }\).
   2. Sketch the curve \(y = - 3 ^ { x }\).