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

6 Show that the solution of the equation $$5 ^ { x } = 3 ^ { x + 4 }$$ can be written as $$x = \frac { \ln 81 } { \ln 5 - \ln 3 }$$ Fully justify your answer.

9 Given that $$\log _ { 2 } x ^ { 3 } - \log _ { 2 } y ^ { 2 } = 9$$ show that $$x = A y ^ { p }$$ where \(A\) is an integer and \(p\) is a rational number. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-15_2488_1716_219_153}

7 In this question you must show detailed reasoning.

1. Express \(\ln 3 \times \ln 9 \times \ln 27\) in terms of \(\ln 3\).
2. Hence show that \(\ln 3 \times \ln 9 \times \ln 27 > 6\).

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Using the laws of logarithms, solve $$\log _ { 4 } ( 12 - 2 x ) = 2 + 2 \log _ { 4 } ( x + 1 )$$

1. The number of bacteria on a surface is being monitored.

The number of bacteria, \(N\), on the surface, \(t\) hours after monitoring began is modelled by the equation $$\log _ { 10 } N = 0.35 t + 2$$ Use the equation of the model to answer parts (a) to (c).

1. Find the initial number of bacteria on the surface.
2. Show that the equation of the model can be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(b\) to 2 decimal places.
3. Hence find the rate of growth of bacteria on the surface exactly 5 hours after monitoring began.

1

1. Express each of the following as a single logarithm.
   1. \(\log _ { a } 5 + \log _ { a } 3\)
   2. \(5 \log _ { b } 2 - 3 \log _ { b } 4\)
2. Express \(\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an algebraic fraction in its simplest form.

7

1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
2. Hence show that \(\int _ { 2 } ^ { 5 } \frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) } \mathrm { d } x = \ln 24\).

1

1. Express each of the following as a single logarithm.
   1. \(\log _ { a } 5 + \log _ { a } 3\)
   2. \(5 \log _ { b } 2 - 3 \log _ { b } 4\)
2. Express \(\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an algebraic fraction in its simplest form.
3. Show that \(\frac { 3 \sqrt { 3 } - 1 } { 2 \sqrt { 3 } - 3 } = \frac { 15 + 7 \sqrt { 3 } } { 3 }\).

2 Express each of the following as a single logarithm.

1. \(\log 3 + \log 4 - \log 2\)
2. \(2 \log x - 3 \log y + 2 \log z\)

3 Solve the equation \(6 ^ { 2 x - 1 } = 3 ^ { x + 2 }\), giving your answer in the form \(x = \frac { \ln a } { \ln b }\) where \(a\) and \(b\) are integers.

2 You are given that \(\ln ( 12 ) = 2.484907\) and \(\ln ( 3 ) = 1.098612\), correct to 6 decimal places. Use the laws of logarithms to obtain the values of \(\ln ( 36 )\) and \(\ln ( 0.5 )\), correct to 4 decimal places. You must show your numerical working.

Use logarithms to solve the equation \(6^{2x-1} = 5e^{3x+2}\). Give your answer correct to 4 significant figures. [4]

Use logarithms to solve the equation \(3^{4t+3} = 5^{2t+7}\). Give your answer correct to 3 significant figures. [4]

The variables \(x\) and \(y\) satisfy the equation \(a^{2y} = e^{3x+k}\), 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 \(\frac{3}{2\ln a}\). [1]
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 \(k\). [4]

\includegraphics{figure_2} Two variable quantities \(x\) and \(y\) are related by the equation $$y = k(a^{-x}),$$ where \(a\) and \(k\) are constants. Four pairs of values of \(x\) and \(y\) are measured experimentally. The result of plotting \(\ln y\) against \(x\) is shown in the diagram. Use the diagram to estimate the values of \(a\) and \(k\). [5]

The polynomial \(\mathrm{p}(x)\) is defined by $$\mathrm{p}(x) = ax^3 + 3x^2 + 4ax - 5,$$ where \(a\) is a constant. It is given that \((2x - 1)\) is a factor of \(\mathrm{p}(x)\).

1. Use the factor theorem to find the value of \(a\). [2]
2. Factorise \(\mathrm{p}(x)\) and hence show that the equation \(\mathrm{p}(x) = 0\) has only one real root. [4]
3. Use logarithms to solve the equation \(\mathrm{p}(6^x) = 0\) correct to 3 significant figures. [2]

Show that \(\int_1^7 \frac{6}{2x + 1} \, dx = \ln 125\). [5]

Solve the equation \(\ln(x^3 - 3) = 3 \ln x - \ln 3\). Give your answer correct to 3 significant figures. [3]

Given that \(x = 4(3^{-y})\), express \(y\) in terms of \(x\). [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]

Find the exact value of \(\int_1^4 \frac{\ln x}{\sqrt{x}} dx\). [5]