Linear relationship between log variables

3. \begin{figure}[h] \end{figure} Figure 1 shows a linear relationship between \(\log _ { 10 } y\) and \(\log _ { 10 } x\) The line passes through the points \(( 0,4 )\) and \(( 6,0 )\) as shown.

1. Find an equation linking \(\log _ { 10 } y\) with \(\log _ { 10 } x\)
2. Hence, or otherwise, express \(y\) in the form \(p x ^ { q }\), where \(p\) and \(q\) are constants to be found.

2. \begin{figure}[h] \end{figure} Figure 1 shows the linear relationship between \(\log _ { 6 } T\) and \(\log _ { 6 } x\) The line passes through the points \(( 0,4 )\) and \(( 2,0 )\) as shown.

1. 1. Find an equation linking \(\log _ { 6 } T\) and \(\log _ { 6 } x\)
   2. Hence find the exact value of \(T\) when \(x = 216\)
2. Find an equation, not involving logs, linking \(T\) with \(x\)

1. (i) The variables \(x\) and \(y\) are connected by the equation

$$y = \frac { 10 ^ { 6 } } { x ^ { 3 } } \quad x > 0$$ Sketch the graph of \(\log _ { 10 } y\) against \(\log _ { 10 } x\) Show on your sketch the coordinates of the points of intersection of the graph with the axes.
(ii) \begin{figure}[h] \end{figure} Figure 2 shows the linear relationship between \(\log _ { 3 } N\) and \(t\).
Show that \(N = a b ^ { t }\) where \(a\) and \(b\) are constants to be found.

The variables \(x\) and \(y\) satisfy the equation \(a^{2y} = e^{3x+k}\), where \(a\) and \(k\) are constants. The graph of \(y\) against \(x\) is a straight line.

1. Use logarithms to show that the gradient of the straight line is \(\frac{3}{2\ln a}\). [1]
2. Given that the straight line passes through the points \((0.4, 0.95)\) and \((3.3, 3.80)\), find the values of \(a\) and \(k\). [4]