y vs ln(x) linear graph

3 \includegraphics[max width=\textwidth, alt={}, center]{83d0697c-b133-47da-a745-dfdafa7dbf10-05_604_933_258_605} The variables \(x\) and \(y\) satisfy the equation \(a ^ { y } = k x\), where \(a\) and \(k\) are constants. The graph of \(y\) against \(\ln x\) is a straight line passing through the points \(( 1.03,6.36 )\) and \(( 2.58,9.00 )\), as shown in the diagram. Find the values of \(a\) and \(k\), giving each value correct to 2 significant figures.

3 \includegraphics[max width=\textwidth, alt={}, center]{6294c4f4-70a9-4b81-87e0-20e2cc24dd27-05_606_933_258_605} The variables \(x\) and \(y\) satisfy the equation \(a ^ { y } = k x\), where \(a\) and \(k\) are constants. The graph of \(y\) against \(\ln x\) is a straight line passing through the points \(( 1.03,6.36 )\) and \(( 2.58,9.00 )\), as shown in the diagram. Find the values of \(a\) and \(k\), giving each value correct to 2 significant figures.

3 The variables \(x\) and \(y\) satisfy the equation \(x = A \left( 3 ^ { - y } \right)\), where \(A\) is a constant.

1. Explain why the graph of \(y\) against \(\ln x\) is a straight line and state the exact value of the gradient of the line.
   It is given that the line intersects the \(y\)-axis at the point where \(y = 1.3\).
2. Calculate the value of \(A\), giving your answer correct to 2 decimal places.

3 \includegraphics[max width=\textwidth, alt={}, center]{37f00894-e6b1-4961-bd3c-4852e43173d0-04_597_921_260_573} The variables \(x\) and \(y\) satisfy the equation \(\mathrm { a } ^ { \mathrm { y } } = \mathrm { bx }\), where \(a\) and \(b\) are constants. The graph of \(y\) against \(\ln x\) is a straight line passing through the points ( \(0.336,1.00\) ) and ( \(1.31,1.50\) ), as shown in the diagram. Find the values of \(a\) and \(b\). Give each value correct to the nearest integer.