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.

A zookeeper models the median mass of infant monkeys born at their zoo, up to the age of 2 years, by the formula $$y = a + b \log_{10} x$$ where \(y\) is the median mass in kilograms, \(x\) is age in months and \(a\) and \(b\) are constants. The zookeeper uses the data shown below to determine the values of \(a\) and \(b\).

Age in months (\(x\))	3	24
Median mass (\(y\))	6.4	12

1. The zookeeper uses the data for monkeys aged 3 months to write the correct equation $$6.4 = a + b \log_{10} 3$$
   1. Use the data for monkeys aged 24 months to write a second equation. [1 mark]
   2. Show that $$b = \frac{5.6}{\log_{10} 8}$$ [3 marks]
   3. Find the value of \(a\). Give your answer to two decimal places. [1 mark]
2. Use a suitable value for \(x\) to determine whether the model can be used to predict the median mass of monkeys less than one week old. [2 marks]