1.06f Laws of logarithms: addition, subtraction, power rules

1 Solve the equation $$\ln ( x + 1 ) - \ln x = 2 \ln 2$$

1

1. Solve the equation \(\ln ( 2 + x ) - \ln x = 2 \ln 3\).
2. Hence solve the equation \(\ln ( 2 + \cot y ) - \ln ( \cot y ) = 2 \ln 3\) for \(0 < y < \frac { 1 } { 2 } \pi\). Give your answer correct to 4 significant figures.

1 Use logarithms to solve the equation \(12 ^ { x } = 3 ^ { 2 x + 1 }\). Give your answer correct to 3 significant figures.

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.

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1. Given that \(2 \ln ( x + 1 ) + \ln x = \ln ( x + 9 )\), show that \(x = \sqrt { \frac { 9 } { x + 2 } }\).
2. It is given that the equation \(x = \sqrt { \frac { 9 } { x + 2 } }\) has a single root. Show by calculation that this root lies between 1.5 and 2.0.
3. Use an iterative formula, based on the equation in part (b), to find the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

1 Given that $$\ln ( 2 x + 1 ) - \ln ( x - 3 ) = 2$$ find \(x\) in terms of e.

2 Given that \(\frac { 2 ^ { 3 x + 2 } + 8 } { 2 ^ { 3 x } - 7 } = 5\), find the value of \(2 ^ { 3 x }\) and hence, using logarithms, find the value of \(x\) correct to 4 significant figures.

4

1. Sketch, on the same diagram, the graphs of \(y = | 3 - x |\) and \(y = 9 - 2 x\).
2. Solve the inequality \(| 3 - x | > 9 - 2 x\).
3. Use logarithms to solve the inequality \(2 ^ { 3 x - 10 } < 500\). Give your answer in the form \(x < a\), where the value of \(a\) is given correct to 3 significant figures.
4. List the integers that satisfy both of the inequalities \(| 3 - x | > 9 - 2 x\) and \(2 ^ { 3 x - 10 } < 500\).

1 The variables \(x\) and \(y\) satisfy the equation \(a ^ { 2 y } = \mathrm { e } ^ { 3 x + k }\), 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 \(\frac { 3 } { 2 \ln a }\).
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 \(k\). \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-03_2723_33_99_21}

1 Use logarithms to show that the equation \(5 ^ { 8 y } = 6 ^ { 7 x }\) can be expressed in the form \(y = k x\). Give the value of the constant \(k\) correct to 3 significant figures.

4

1. Solve the equation \(5 ^ { 2 x } + 5 ^ { x } = 12\), giving your answer correct to 3 significant figures.
2. It is given that \(\ln ( y + 5 ) - \ln y = 2 \ln x\). Express \(y\) in terms of \(x\), in a form not involving logarithms.

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 \(n\).

1 Given that \(2 ^ { x } = 5 ^ { y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 3 significant figures.

2

1. Use logarithms to solve the equation \(3 ^ { X } = 8\), giving your answer correct to 2 decimal places.
2. It is given that $$\ln z = \ln ( y + 2 ) - 2 \ln y$$ where \(y > 0\). Express \(z\) in terms of \(y\) in a form not involving logarithms.

2 The variables \(x\) and \(y\) satisfy the relation \(3 ^ { y } = 4 ^ { x + 2 }\).

1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. Find the exact value of the gradient of this line.
2. Calculate the \(x\)-coordinate of the point of intersection of this line with the line \(y = 2 x\), giving your answer correct to 2 decimal places.

2 Use logarithms to solve the equation \(4 ^ { x } = 2 \left( 3 ^ { x } \right)\), giving your answer correct to 3 significant figures.

1 Given that \(( 1.25 ) ^ { x } = ( 2.5 ) ^ { y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 3 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{b9556031-871d-4dd3-9523-e3438a41339f-3_655_685_262_730} The diagram shows the curve \(y = x \mathrm { e } ^ { 2 x }\) and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Show that the curve intersects the line \(y = 20\) at the point whose \(x\)-coordinate is the root of the equation $$x = \frac { 1 } { 2 } \ln \left( \frac { 20 } { x } \right)$$
3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 20 } { x _ { n } } \right)$$ with initial value \(x _ { 1 } = 1.3\), to calculate the root correct to 2 decimal places, giving the result of each iteration to 4 decimal places.

2 Show that \(\int _ { 0 } ^ { 6 } \frac { 1 } { x + 2 } \mathrm {~d} x = 2 \ln 2\).

1 Given that \(13 ^ { x } = ( 2.8 ) ^ { y }\), use logarithms to show that \(y = k x\) and find the value of \(k\) correct to 3 significant figures.

1 Use logarithms to solve the equation \(3 ^ { x } = 2 ^ { x + 2 }\), giving your answer correct to 3 significant figures.

2

1. Given that \(5 ^ { 2 x } + 5 ^ { x } = 12\), find the value of \(5 ^ { x }\).
2. Hence, using logarithms, solve the equation \(5 ^ { 2 x } + 5 ^ { x } = 12\), giving the value of \(x\) correct to 3 significant figures.

2 Solve the equation \(\ln ( 3 - 2 x ) - 2 \ln x = \ln 5\).

4 The variables \(x\) and \(y\) satisfy the equation \(5 ^ { y + 1 } = 2 ^ { 3 x }\).

1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line.
2. Find the exact value of the gradient of this line and state the coordinates of the point at which the line cuts the \(y\)-axis.