Simultaneous equations with logarithms

2 \log _ { 2 } y = 5 - \log _ { 2 } x
\log _ { x } y = - 3 \end{gathered}$$ for \(x > 0 , y > 0\)

1. (a) Show that the equation

$$2 \log _ { 2 } y = 5 - \log _ { 2 } x \quad x > 0 , y > 0$$ may be written in the form \(y ^ { 2 } = \frac { k } { x }\) where \(k\) is a constant to be found.
(b) Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} 2 \log _ { 2 } y = 5 - \log _ { 2 } x \\ \log _ { x } y = - 3 \end{gathered}$$ for \(x > 0 , y > 0\)

4. In this question you must show all stages of your working. Give your answers in fully simplified surd form. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations $$\begin{gathered} a - b = 8 \\ \log _ { 4 } a + \log _ { 4 } b = 3 \end{gathered}$$ (6)

1. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations

$$\begin{gathered} a = 3 b , \\ \log _ { 3 } a + \log _ { 3 } b = 2 . \end{gathered}$$ Give your answers as exact numbers. \section*{6.} \begin{figure}[h] \end{figure} Figure 1 shows 3 yachts \(A , B\) and \(C\) which are assumed to be in the same horizontal plane. Yacht \(B\) is 500 m due north of yacht \(A\) and yacht \(C\) is 700 m from \(A\). The bearing of \(C\) from \(A\) is \(015 ^ { \circ }\).

1. Calculate the distance between yacht \(B\) and yacht \(C\), in metres to 3 significant figures. The bearing of yacht \(C\) from yacht \(B\) is \(\theta ^ { \circ }\), as shown in Figure 1.
2. Calculate the value of \(\theta\).

1. (i) Given that

$$\log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x$$ show that $$y = 2 x + 1$$ (ii) Solve the simultaneous equations $$\begin{aligned} & \log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x \\ & 2 \log _ { 3 } y = 2 + \log _ { 3 } x \end{aligned}$$

8. (a) Given that $$\log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x ,$$ show that $$y = 2 x + 1 .$$ (b) Solve the simultaneous equations $$\begin{aligned} & \log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x \\ & 2 \log _ { 3 } y = 2 + \log _ { 3 } x \end{aligned}$$