1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form

4 \includegraphics[max width=\textwidth, alt={}, center]{01f2de2b-3482-4694-889e-7fcd016b57e3-06_659_828_262_660} The variables \(x\) and \(y\) satisfy the equation \(y = A x ^ { - 2 p }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( - 0.68,3.02 )\) and \(( 1.07 , - 1.53 )\), as shown in the diagram. Find the values of \(A\) and \(p\).

4 \includegraphics[max width=\textwidth, alt={}, center]{ad833f8c-80de-42ae-a186-93091a6fdf1e-06_659_828_262_660} The variables \(x\) and \(y\) satisfy the equation \(y = A x ^ { - 2 p }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( - 0.68,3.02 )\) and \(( 1.07 , - 1.53 )\), as shown in the diagram. Find the values of \(A\) and \(p\).

1 \includegraphics[max width=\textwidth, alt={}, center]{ed12a4fb-e3bf-4d00-ad09-9ba5be941dd5-02_654_396_258_872} The variables \(x\) and \(y\) satisfy the equation \(y = 4 ^ { 2 x - a }\), where \(a\) is an integer. As shown in the diagram, the graph of \(\ln y\) against \(x\) is a straight line passing through the point \(( 0 , - 20.8 )\), where the second coordinate is given correct to 3 significant figures.

1. Show that the gradient of the straight line is \(\ln 16\).
2. Determine the value of \(a\).

2 \includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-03_515_598_260_762} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { ( A - B ) x }\), where \(A\) and \(B\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.4,3.6 )\) and \(( 2.9,14.1 )\), as shown in the diagram. Find the values of \(A\) and \(B\) correct to 3 significant figures.

2 \includegraphics[max width=\textwidth, alt={}, center]{a1ea242a-c7f4-46b0-b4b8-bd13b3880557-03_515_598_260_762} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { ( A - B ) x }\), where \(A\) and \(B\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.4,3.6 )\) and \(( 2.9,14.1 )\), as shown in the diagram. Find the values of \(A\) and \(B\) correct to 3 significant figures.

3 The variables \(x\) and \(y\) satisfy the equation \(y = 3 ^ { 2 a } a ^ { x }\), where \(a\) is a constant. The graph of \(\ln y\) against \(x\) is a straight line with gradient 0.239 .

1. Find the value of \(a\) correct to 3 significant figures.
2. Hence find the value of \(x\) when \(y = 36\). Give your answer correct to 3 significant figures.

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 \includegraphics[max width=\textwidth, alt={}, center]{68f4b2dc-a05d-4061-aaf0-de15cfe186a9-04_714_515_262_804} The variables \(x\) and \(y\) satisfy the equation \(y = A x ^ { k }\), where \(A\) and \(k\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points ( \(0.56,2.87\) ) and ( \(0.81,3.47\) ), as shown in the diagram. Find the value of \(k\), and the value of \(A\) correct to 2 significant figures.

3 Two variable quantities \(x\) and \(y\) are related by the equation $$y = A x ^ { n }$$ where \(A\) and \(n\) are constants. \includegraphics[max width=\textwidth, alt={}, center]{9b103197-7ba0-427a-b983-34edb51b6cca-2_422_697_977_740} When a graph is plotted showing values of \(\ln y\) on the vertical axis and values of \(\ln x\) on the horizontal axis, the points lie on a straight line. This line crosses the vertical axis at the point ( \(0,2.3\) ) and also passes through the point (4.0,1.7), as shown in the diagram. Find the values of \(A\) and \(n\).

3 \includegraphics[max width=\textwidth, alt={}, center]{d90dc270-b304-4b42-8e0e-37641b8a03b8-2_556_1113_680_516} The variables \(x\) and \(y\) satisfy the equation \(y = K x ^ { m }\), where \(K\) and \(m\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( 0,2.0 )\) and \(( 6,10.2 )\), as shown in the diagram. Find the values of \(K\) and \(m\), correct to 2 decimal places.

2 \includegraphics[max width=\textwidth, alt={}, center]{beb8df77-e091-4248-812b-20e885c42e37-2_453_771_386_685} The variables \(x\) and \(y\) satisfy the equation \(y = A \left( b ^ { x } \right)\), where \(A\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0,2.14 )\) and \(( 5,4.49 )\), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 1 decimal place.

2 \includegraphics[max width=\textwidth, alt={}, center]{0a45a806-007f-4840-85e7-16d4c1a2c599-2_453_771_386_685} The variables \(x\) and \(y\) satisfy the equation \(y = A \left( b ^ { x } \right)\), where \(A\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0,2.14 )\) and \(( 5,4.49 )\), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 1 decimal place.

5 \includegraphics[max width=\textwidth, alt={}, center]{de8af872-9f77-4787-8e66-ed199405ca25-2_583_597_1457_772} The variables \(x\) and \(y\) satisfy the equation \(y = K \left( 2 ^ { p x } \right)\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(1.35,1.87\) ) and ( \(3.35,3.81\) ), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places.
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5 \includegraphics[max width=\textwidth, alt={}, center]{22ba6cc7-7375-434e-9eaa-d536684dd727-2_583_597_1457_772} The variables \(x\) and \(y\) satisfy the equation \(y = K \left( 2 ^ { p x } \right)\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(1.35,1.87\) ) and ( \(3.35,3.81\) ), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places.
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2 \includegraphics[max width=\textwidth, alt={}, center]{595e38f4-c52e-4509-8b16-f08e30dec96b-2_456_716_529_712} The variables \(x\) and \(y\) satisfy the equation $$y = A \mathrm { e } ^ { p ( x - 1 ) } ,$$ where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 2,1.60 )\) and \(( 5,2.92 )\), as shown in the diagram. Find the values of \(A\) and \(p\) correct to 2 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{de2f8bf3-fd03-4199-9eb2-c9cbac4d4385-05_551_535_260_806} The variables \(x\) and \(y\) satisfy the equation \(y = \frac { K } { a ^ { 2 x } }\), where \(K\) and \(a\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.6,1.81 )\) and \(( 1.4,1.39 )\), as shown in the diagram. Find the values of \(K\) and \(a\) correct to 2 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{bdc467f6-105e-4429-95c6-701eaa43deff-05_551_533_260_806} The variables \(x\) and \(y\) satisfy the equation \(y = \frac { K } { a ^ { 2 x } }\), where \(K\) and \(a\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.6,1.81 )\) and \(( 1.4,1.39 )\), as shown in the diagram. Find the values of \(K\) and \(a\) correct to 2 significant figures.

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.

3 The variables \(x\) and \(y\) satisfy the equation \(x ^ { n } y = C\), where \(n\) and \(C\) are constants. When \(x = 1.10\), \(y = 5.20\), and when \(x = 3.20 , y = 1.05\).

1. Find the values of \(n\) and \(C\).
2. Explain why the graph of \(\ln y\) against \(\ln x\) is a straight line.

2 \includegraphics[max width=\textwidth, alt={}, center]{9275a3ed-8820-481b-9fc8-28c21b81dbed-2_559_789_513_678} Two variable quantities \(x\) and \(y\) are related by the equation \(y = A x ^ { n }\), where \(A\) and \(n\) are constants. The diagram shows the result of plotting \(\ln y\) against \(\ln x\) for four pairs of values of \(x\) and \(y\). Use the diagram to estimate the values of \(A\) and \(n\).

2 Two variable quantities \(x\) and \(y\) are believed to satisfy an equation of the form \(y = C \left( a ^ { x } \right)\), where \(C\) and \(a\) are constants. An experiment produced four pairs of values of \(x\) and \(y\). The table below gives the corresponding values of \(x\) and \(\ln y\). By plotting \(\ln y\) against \(x\) for these four pairs of values and drawing a suitable straight line, estimate the values of \(C\) and \(a\). Give your answers correct to 2 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{21878d10-7f16-4dbb-86ef-65a7ba5eeafb-03_759_944_749_596}

3 \includegraphics[max width=\textwidth, alt={}, center]{772c14a1-f79a-4147-a293-0ff34f930e20-04_577_569_260_788} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { p x + p }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 1,2.835 )\) and \(( 6,6.585 )\), as shown in the diagram. Find the values of \(A\) and \(p\).

3

1. Solve the inequality \(| y - 5 | < 1\).
2. Hence solve the inequality \(\left| 3 ^ { x } - 5 \right| < 1\), giving 3 significant figures in your answer.

3 \includegraphics[max width=\textwidth, alt={}, center]{733c3711-0429-415d-a8f3-8de86097635a-2_550_843_769_651} The variables \(x\) and \(y\) satisfy the equation \(y = A \left( b ^ { - x } \right)\), where \(A\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(0,1.3\) ) and ( \(1.6,0.9\) ), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 2 decimal places.