Solve by showing reduces to polynomial

2

1. Show that the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$ can be written as a quadratic equation in \(x\).
2. Hence solve the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$

1

1. Show that the equation \(\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x\) can be written as a quadratic equation in \(x\).
2. Hence solve the equation \(\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x\), giving your answer correct to 3 significant figures.

3

1. Show that the equation $$\log _ { 10 } ( x + 5 ) = 2 - \log _ { 10 } x$$ may be written as a quadratic equation in \(x\).
2. Hence find the value of \(x\) satisfying the equation $$\log _ { 10 } ( x + 5 ) = 2 - \log _ { 10 } x$$

3

1. Show that the equation \(\log _ { 3 } ( 2 x + 1 ) = 1 + 2 \log _ { 3 } ( x - 1 )\) can be written as a quadratic equation in \(x\).
2. Hence solve the equation \(\log _ { 3 } ( 4 y + 1 ) = 1 + 2 \log _ { 3 } ( 2 y - 1 )\), giving your answer correct to 2 decimal places.

6. Given that $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$

1. show that $$4 x ^ { 2 } - 16 x - 9 = 0$$
2. Hence solve the equation $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$

9. Given that \(y = 3 x ^ { 2 }\),

1. show that \(\log _ { 3 } y = 1 + 2 \log _ { 3 } x\)
2. Hence, or otherwise, solve the equation $$1 + 2 \log _ { 3 } x = \log _ { 3 } ( 28 x - 9 )$$

1. (a) Given that

$$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$ show that $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$ (b) Given also that - 1 is a root of the equation $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$

1. use algebra to find the other two roots of the equation.
2. Hence solve $$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$

7. (a) Given that $$3 \log _ { 3 } ( 2 x - 1 ) = 2 + \log _ { 3 } ( 14 x - 25 )$$ show that $$2 x ^ { 3 } - 3 x ^ { 2 } - 30 x + 56 = 0$$ (b) Show that - 4 is a root of this cubic equation.
(c) Hence, using algebra and showing each step of your working, solve $$3 \log _ { 3 } ( 2 x - 1 ) = 2 + \log _ { 3 } ( 14 x - 25 )$$

4. Given that \(y = 3 x ^ { 2 }\),

1. show that \(\log _ { 3 } y = 1 + 2 \log _ { 3 } x\)
2. Hence, or otherwise, solve the equation $$1 + 2 \log _ { 3 } x = \log _ { 3 } ( 28 x - 9 )$$

6. Given that $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$

1. Show that $$x ^ { 2 } - 34 x + 225 = 0$$
2. Hence, or otherwise, solve the equation $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$

7. (a) Given that $$2 \log _ { 3 } ( x - 5 ) - \log _ { 3 } ( 2 x - 13 ) = 1$$ show that \(x ^ { 2 } - 16 x + 64 = 0\).
(b) Hence, or otherwise, solve \(2 \log _ { 3 } ( x - 5 ) - \log _ { 3 } ( 2 x - 13 ) = 1\).

8

1. Show that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\), where \(k\) is a constant, can be expressed in the form \(x ^ { 2 } - 8 k x + 8 = 0\).
2. Given that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\) has only one real root, find the value of this root.

7 It is given that \(n\) satisfies the equation $$2 \log _ { a } n - \log _ { a } ( 5 n - 24 ) = \log _ { a } 4$$

1. Show that \(n ^ { 2 } - 20 n + 96 = 0\).
2. Hence find the possible values of \(n\).

8

1. Given that \(\log _ { a } b = c\), express \(b\) in terms of \(a\) and \(c\).
2. By forming a quadratic equation, show that there is only one value of \(x\) which satisfies the equation \(2 \log _ { 2 } ( x + 7 ) - \log _ { 2 } ( x + 5 ) = 3\).

9

1. Given that \(\log _ { 3 } c = m\) and \(\log _ { 27 } d = n\), express \(\frac { \sqrt { c } } { d ^ { 2 } }\) in the form \(3 ^ { y }\), where \(y\) is an expression in terms of \(m\) and \(n\).
2. Show that the equation $$\log _ { 4 } ( 2 x + 3 ) + \log _ { 4 } ( 2 x + 15 ) = 1 + \log _ { 4 } ( 14 x + 5 )$$ has only one solution and state its value.
   [0pt] [4 marks]

5. (a) Given that \(3 + 2 \log _ { 2 } x = \log _ { 2 } y\), show that \(y = 8 x ^ { 2 }\).
(b) Hence, or otherwise, find the roots \(\alpha\) and \(\beta\), where \(\alpha < \beta\), of the equation $$3 + 2 \log _ { 2 } x = \log _ { 2 } ( 14 x - 3 )$$ (c) Show that \(\log _ { 2 } \alpha = - 2\).
(d) Calculate \(\log _ { 2 } \beta\), giving your answer to 3 significant figures.