Solve log equation with domain restrictions

1. Given that

$$\log _ { 2 } ( x + 3 ) + \log _ { 2 } ( x + 10 ) = 2 + 2 \log _ { 2 } x$$

1. show that $$3 x ^ { 2 } - 13 x - 30 = 0$$
2. 1. Write down the roots of the equation $$3 x ^ { 2 } - 13 x - 30 = 0$$
   2. Hence state which of the roots in part (b)(i) is not a solution of $$\log _ { 2 } ( x + 3 ) + \log _ { 2 } ( x + 10 ) = 2 + 2 \log _ { 2 } x$$ giving a reason for your answer.

1. (a) Given that

$$2 \log ( 4 - x ) = \log ( x + 8 )$$ show that $$x ^ { 2 } - 9 x + 8 = 0$$ (b) (i) Write down the roots of the equation $$x ^ { 2 } - 9 x + 8 = 0$$ (ii) State which of the roots in (b)(i) is not a solution of $$2 \log ( 4 - x ) = \log ( x + 8 )$$ giving a reason for your answer.

6 Show that the equation $$\begin{aligned} & \qquad 2 \log _ { 10 } x = \log _ { 10 } 4 + \log _ { 10 } ( x + 8 ) \\ & \text { has exactly one solution. } \\ & \text { Fully justify your answer. } \end{aligned}$$

Solve $$2 \log_3 x - \log_3 (x - 2) = 2, \quad x > 2.$$ [6]

Find the exact solution of the equation \(\ln(x + 1) + \ln(x - 1) = \ln 15 - 2\ln 7\) Fully justify your answer. [5 marks]

Find the values of \(x\) such that $$2\log_3 x - \log_3(x - 2) = 2$$ [5]

Solve the inequality $$\log_3(2x^2 - x) - \log_3(2x^2 - 3x + 1) > 1.$$ [5]