ln(y) vs non-x variable: find constants

Given that ln(y) against a transformed variable (e.g. x², ln(x)) is a straight line, find the constants in the underlying equation such as y = Ae^(-kx²) or y = A×B^(ln x).

3 questions · Moderate -0.3

1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form
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CAIE P2 2018 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{873a104f-e2e2-49bb-b943-583769728fbb-04_554_493_260_826} The variables \(x\) and \(y\) satisfy the equation \(y = A \times B ^ { \ln x }\), where \(A\) and \(B\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points (2.2, 4.908) and (5.9, 11.008), as shown in the diagram. Find the values of \(A\) and \(B\) correct to 2 significant figures.
CAIE P3 2010 June Q2
4 marks Moderate -0.8
The variables \(x\) and \(y\) satisfy the equation \(y^3 = Ae^{2x}\), where \(A\) is a constant. The graph of \(\ln y\) against \(x\) is a straight line.
  1. Find the gradient of this line. [2]
  2. Given that the line intersects the axis of \(\ln y\) at the point where \(\ln y = 0.5\), find the value of \(A\) correct to 2 decimal places. [2]
CAIE P3 2013 June Q3
5 marks Moderate -0.3
\includegraphics{figure_3} The variables \(x\) and \(y\) satisfy the equation \(y = Ae^{-kx^2}\), where \(A\) and \(k\) are constants. The graph of \(\ln y\) against \(x^2\) is a straight line passing through the points \((0.64, 0.76)\) and \((1.69, 0.32)\), as shown in the diagram. Find the values of \(A\) and \(k\) correct to 2 decimal places. [5]