Natural logarithm equation solving

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

2 Solve the equation $$2 \ln \left( 5 - \mathrm { e } ^ { - 2 x } \right) = 1$$ giving your answer correct to 3 significant figures.

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

1 Solve the equation \(5 \ln \left( 4 - 3 ^ { x } \right) = 6\). Show all necessary working and give the answer 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.

9. (i) Find the exact solutions to the equations

1. \(\ln ( 3 x - 7 ) = 5\)
2. \(3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15\) (ii) The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R } \\ \mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , x > 1 \end{array}$$
3. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
4. Find fg and state its range.

Find the exact solutions to the equations

1. \(\ln x + \ln 3 = \ln 6\),
2. \(\mathrm { e } ^ { x } + 3 \mathrm { e } ^ { - x } = 4\).

6. Find algebraically the exact solutions to the equations

1. \(\ln ( 4 - 2 x ) + \ln ( 9 - 3 x ) = 2 \ln ( x + 1 ) , \quad - 1 < x < 2\)
2. \(2 ^ { x } \mathrm { e } ^ { 3 x + 1 } = 10\) Give your answer to (b) in the form \(\frac { a + \ln b } { c + \ln d }\) where \(a , b , c\) and \(d\) are integers.

2. Find the exact solutions, in their simplest form, to the equations

1. \(2 \ln ( 2 x + 1 ) - 10 = 0\)
2. \(3 ^ { x } \mathrm { e } ^ { 4 x } = \mathrm { e } ^ { 7 }\)

2. Find the exact solutions, in their simplest form, to the equations

1. \(\mathrm { e } ^ { 3 x - 9 } = 8\)
2. \(\ln ( 2 y + 5 ) = 2 + \ln ( 4 - y )\) \includegraphics[max width=\textwidth, alt={}, center]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-05_37_1813_0_6}

3. Find the exact solutions of

1. \(\mathrm { e } ^ { 2 x + 3 } = 6\),
2. \(\ln ( 3 x + 2 ) = 4\).

2. (i) Solve the equation $$\ln ( 3 x + 1 ) = 2$$ giving your answer in terms of e.
(ii) Prove, by counter-example, that the statement $$\text { "ln } \left( 3 x ^ { 2 } + 5 x + 3 \right) \geq 0 \text { for all real values of } x \text { " }$$ is false.

2. Solve each equation, giving your answers in exact form.

1. \(\quad \ln ( 2 x - 3 ) = 1\)
2. \(3 \mathrm { e } ^ { y } + 5 \mathrm { e } ^ { - y } = 16\)

5. The function \(f\) is defined by $$\mathrm { f } ( x ) \equiv 4 - \ln 3 x , \quad x \in \mathbb { R } , \quad x > 0$$

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Sketch the curve \(y = \mathrm { f } ( x )\). The function g is defined by $$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
3. Show that $$\operatorname { fg } ( x ) = x + a - \ln b$$ where \(a\) and \(b\) are integers to be found.

5

1. Given that \(2 \ln x = 5\), find the exact value of \(x\).
2. Solve the equation $$2 \ln x + \frac { 15 } { \ln x } = 11$$ giving your answers as exact values of \(x\).

6

1. Given that \(3 \ln x = 4\), find the exact value of \(x\).
2. By forming a quadratic equation in \(\ln x\), solve \(3 \ln x + \frac { 20 } { \ln x } = 19\), giving your answers for \(x\) in an exact form.

3. Giving your answers to 2 decimal places, solve the simultaneous equations $$\begin{aligned} & \mathrm { e } ^ { 2 y } - x + 2 = 0 \\ & \ln ( x + 3 ) - 2 y - 1 = 0 \end{aligned}$$

1. (a) Simplify

$$\frac { x ^ { 2 } + 7 x + 12 } { 2 x ^ { 2 } + 9 x + 4 }$$ (b) Solve the equation $$\ln \left( x ^ { 2 } + 7 x + 12 \right) - 1 = \ln \left( 2 x ^ { 2 } + 9 x + 4 \right)$$ giving your answer in terms of e.

3. Solve each equation, giving your answers in exact form.

1. \(\quad \ln ( 2 x - 3 ) = 1\)
2. \(3 \mathrm { e } ^ { y } + 5 \mathrm { e } ^ { - y } = 16\)

6. The function f is defined by $$\mathrm { f } ( x ) \equiv 4 - \ln 3 x , \quad x \in \mathbb { R } , \quad x > 0 .$$

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Sketch the curve \(y = \mathrm { f } ( x )\).
3. Find an expression for the inverse function, \(\mathrm { f } ^ { - 1 } ( x )\). The function \(g\) is defined by $$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
4. Show that $$\operatorname { fg } ( x ) = x + a - \ln b$$ where \(a\) and \(b\) are integers to be found.

3. (a) Given that \(y = \ln x\), find expressions in terms of \(y\) for

1. \(\quad \log _ { 2 } x\),
2. \(\ln \frac { x ^ { 2 } } { \mathrm { e } }\).
   (b) Hence, or otherwise, solve the equation $$\log _ { 2 } x = 4 - \ln \frac { x ^ { 2 } } { \mathrm { e } } ,$$ giving your answer to 2 decimal places.

3. (a) Solve the equation $$\ln ( 3 x + 1 ) = 2$$ giving your answer in terms of e.
(b) Prove, by counter-example, that the statement $$\text { "ln } \left( 3 x ^ { 2 } + 5 x + 3 \right) \geq 0 \text { for all real values of } x \text { " }$$ is false.

3. (a) Simplify $$\frac { 2 x ^ { 2 } + 3 x - 9 } { 2 x ^ { 2 } - 7 x + 6 }$$ (b) Solve the equation $$\ln \left( 2 x ^ { 2 } + 3 x - 9 \right) = 2 + \ln \left( 2 x ^ { 2 } - 7 x + 6 \right)$$ giving your answer in terms of e.

4 In this question you must show detailed reasoning. Solve the simultaneous equations \(\mathrm { e } ^ { x } - 2 \mathrm { e } ^ { y } = 3\) \(\mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { 2 y } = 33\). Give your answer in an exact form.