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

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

6

1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin x ( 4 \sin x + 6 \cos x ) \mathrm { d } x\).
2. Given that \(\int _ { 0 } ^ { a } \frac { 6 } { 3 x + 2 } \mathrm {~d} x = \ln 49\), find the value of the positive constant \(a\).

1 Solve the equation \(\ln ( 3 x + 1 ) - \ln ( x + 2 ) = 1\), giving your answer in terms of e.

1

1. Solve the equation \(| 9 x - 2 | = | 3 x + 2 |\).
2. Hence, using logarithms, solve the equation \(\left| 3 ^ { y + 2 } - 2 \right| = \left| 3 ^ { y + 1 } + 2 \right|\), giving your answer correct to 3 significant figures.

2 Given that \(9 ^ { x } + 3 ^ { x } = 240\), find the value of \(3 ^ { x }\) and hence, using logarithms, find the value of \(x\) correct to 4 significant figures.

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

1 Find the set of values of \(x\) for which \(2 \left( 3 ^ { 1 - 2 x } \right) < 5 ^ { x }\). Give your answer in a simplified exact form. [4]

3

1. Show that the equation $$\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0$$ can be expressed as a quadratic equation in \(\mathrm { e } ^ { x }\).
2. Hence solve the equation \(\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0\), giving your answer correct to 3 decimal places.

2 Find the real root of the equation \(\frac { 2 \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { 2 + \mathrm { e } ^ { x } } = 3\), giving your answer correct to 3 decimal places. Your working should show clearly that the equation has only one real root.

3 The variables \(x\) and \(y\) satisfy the equation \(x = A \left( 3 ^ { - y } \right)\), where \(A\) is a constant.

1. Explain why the graph of \(y\) against \(\ln x\) is a straight line and state the exact value of the gradient of the line.
   It is given that the line intersects the \(y\)-axis at the point where \(y = 1.3\).
2. Calculate the value of \(A\), giving your answer correct to 2 decimal places.

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

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 \(\ln \left( \mathrm { e } ^ { 2 x } + 3 \right) = 2 x + \ln 3\), giving your answer correct to 3 decimal places.

1 Solve the equation $$3 \mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { - 2 x } = 5$$ Give the answer correct to 3 decimal places.

2 Solve the equation \(\ln \left( 2 x ^ { 2 } - 3 \right) = 2 \ln x - \ln 2\), giving your answer in an exact form.

1 Solve the equation \(\ln ( x + 5 ) = 5 + \ln x\). Give your answer correct to 3 decimal places.

2 Solve the equation \(\ln ( x - 5 ) = 7 - \ln x\). Give your answer correct to 2 decimal places.

4 Solve the equation $$\log _ { 10 } ( 2 x + 1 ) = 2 \log _ { 10 } ( x + 1 ) - 1$$ Give your answers correct to 3 decimal places.

1 Solve the equation $$\ln \left( 1 + \mathrm { e } ^ { - 3 x } \right) = 2$$ Give the answer correct to 3 decimal places.

1 Solve the equation \(4 \left| 5 ^ { x } - 1 \right| = 5 ^ { x }\), giving your answers correct to 3 decimal places.

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.

8 In a certain chemical reaction the amount, \(x\) grams, of a substance is increasing. The differential equation satisfied by \(x\) and \(t\), the time in seconds since the reaction began, is $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k x \mathrm { e } ^ { - 0.1 t }$$ where \(k\) is a positive constant. It is given that \(x = 20\) at the start of the reaction.

1. Solve the differential equation, obtaining a relation between \(x , t\) and \(k\).
2. Given that \(x = 40\) when \(t = 10\), find the value of \(k\) and find the value approached by \(x\) as \(t\) becomes large.

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.