Solve log equation reducing to quadratic

2 Showing all necessary working, solve the equation \(2 \log _ { 2 } x = 3 + \log _ { 2 } ( x + 1 )\), giving your answer correct to 3 significant figures.

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

6. Find the exact values of \(x\) for which $$2 \log _ { 5 } ( x + 5 ) - \log _ { 5 } ( 2 x + 2 ) = 2$$ Give your answers as simplified surds.

4. Using the laws of logarithms, solve $$\log _ { 3 } ( 32 - 12 x ) = 2 \log _ { 3 } ( 1 - x ) + 3$$

1. Use the laws of logarithms to solve

$$\log _ { 2 } ( 16 x ) + \log _ { 2 } ( x + 1 ) = 3 + \log _ { 2 } ( x + 6 )$$

4. Given that \(0 < x < 4\) and $$\log _ { 5 } ( 4 - x ) - 2 \log _ { 5 } x = 1$$ find the value of \(x\).
(6)

2. Find the values of \(x\) such that $$2 \log _ { 3 } x - \log _ { 3 } ( x - 2 ) = 2$$

1.(a)Solve the equation $$\sqrt { } ( 3 x + 16 ) = 3 + \sqrt { } ( x + 1 )$$ (b)Solve the equation $$\log _ { 3 } ( x - 7 ) - \frac { 1 } { 2 } \log _ { 3 } x = 1 - \log _ { 3 } 2$$

1. Using the laws of logarithms, solve the equation

$$2 \log _ { 5 } ( 3 x - 2 ) - \log _ { 5 } x = 2$$

1. Find any real values of \(x\) such that

$$2 \log _ { 4 } ( 2 - x ) - \log _ { 4 } ( x + 5 ) = 1$$

3. Use the laws of logarithms to solve the equation $$2 + \log _ { 2 } ( 2 x + 1 ) = 2 \log _ { 2 } ( 22 - x )$$

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Using the laws of logarithms, solve $$\log _ { 4 } ( 12 - 2 x ) = 2 + 2 \log _ { 4 } ( x + 1 )$$

Given that the equation $$2\log_2 x = \log_2(kx - 1) + 3,$$ only has one solution, find the value of \(x\). [6]