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

1. Using the laws of logarithms, solve the equation

$$2 \log _ { 5 } ( 3 x - 2 ) - \log _ { 5 } x = 2$$

9. $$\begin{aligned} p & = \log _ { a } 16 \\ q & = \log _ { a } 25 \end{aligned}$$ where \(a\) is a constant.
Find in terms of \(p\) and/or \(q\),

1. \(\log _ { a } 256\)
2. \(\log _ { a } 100\)
3. \(\log _ { a } 80 \times \log _ { a } 3.2\)

1. Find any real values of \(x\) such that

$$2 \log _ { 4 } ( 2 - x ) - \log _ { 4 } ( x + 5 ) = 1$$

4. a. Express \(\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x \quad\) as an integral.
b. Hence show that $$\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x = \ln k$$ where \(k\) is a constant to be found.

3. Use the laws of logarithms to solve the equation $$2 + \log _ { 2 } ( 2 x + 1 ) = 2 \log _ { 2 } ( 22 - x )$$

2. Solve $$4 ^ { x - 3 } = 6$$ giving your answer in the form \(a + b \log _ { 2 } 3\), where \(a\) and \(b\) are constants to be found.

1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 3 } ( x )\) The values of \(y\) are given to 2 decimal places as appropriate.

a. Obtain an estimate for \(\int _ { 3 } ^ { 9 } \log _ { 3 } ( x ) \mathrm { d } x\), giving your answer to two decimal places. Use your answer to part (a) and making your method clear, estimate
b. i) \(\int _ { 3 } ^ { 9 } \log _ { 3 } \sqrt { x } \mathrm {~d} x\) ii) \(\int _ { 3 } ^ { 18 } \log _ { 3 } \left( 9 x ^ { 3 } \right) \mathrm { d } x\)

14. a. Express \(\frac { 1 } { ( 3 - x ) ( 1 - x ) }\) in partial fractions.
(2) A scientist is studying the mass of a substance in a laboratory.
The mass, \(x\) grams, of a substance at time \(t\) seconds after a chemical reaction starts is modelled by the differential equation $$2 \frac { d x } { d t } = ( 3 - x ) ( 1 - x ) \quad t \geq 0,0 \leq x < 1$$ Given that when \(t = 0 , x = 0\) b. solve the differential equation and show that the solution can be written as $$x = \frac { 3 \left( e ^ { t } - 1 \right) } { 3 e ^ { t } - 1 }$$ c. Find the mass, \(x\) grams, which has formed 2 seconds after the start of the reaction. Give your answer correct to 3 significant figures.
d. Find the limiting value of \(x\) as \(t\) increases.

1. Given that \(a > b > 0\) and that \(a\) and \(b\) satisfy the equation

$$\log a - \log b = \log ( a - b )$$

1. show that $$a = \frac { b ^ { 2 } } { b - 1 }$$ (3)
2. Write down the full restriction on the value of \(b\), explaining the reason for this restriction.

6. $$a = \log _ { 2 } x \quad b = \log _ { 2 } ( x + 8 )$$ Express in terms of \(a\) and/or \(b\)

1. \(\log _ { 2 } \sqrt { x }\)
2. \(\log _ { 2 } \left( x ^ { 2 } + 8 x \right)\)
3. \(\log _ { 2 } \left( 8 + \frac { 64 } { x } \right)\) Give your answer in simplest form.

3. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 4 x + 10 y = k$$ where \(k\) is a constant.

1. Find the coordinates of the centre of \(C\).
2. State the range of possible values for \(k\).

1. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 48\), where \(a\) is a constant

Given that \(\mathrm { f } ( - 6 ) = 0\)

1. 1. show that \(a = 4\)
   2. express \(\mathrm { f } ( x )\) as a product of two algebraic factors. Given that \(2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3\)
2. show that \(x ^ { 3 } + 4 x ^ { 2 } - 4 x + 48 = 0\)
3. hence explain why $$2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3$$ has no real roots.

1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 3 } 2 x\) The values of \(y\) are given to 2 decimal places as appropriate.

1. Using the trapezium rule with all the values of \(y\) in the table, find an estimate for $$\int _ { 3 } ^ { 9 } \log _ { 3 } 2 x \mathrm {~d} x$$ Using your answer to part (a) and making your method clear, estimate
2. 1. \(\int _ { 3 } ^ { 9 } \log _ { 3 } ( 2 x ) ^ { 10 } \mathrm {~d} x\)
   2. \(\int _ { 3 } ^ { 9 } \log _ { 3 } 18 x \mathrm {~d} x\)

1. Given that

$$\log _ { 2 } ( x + 3 ) + \log _ { 2 } ( x + 10 ) = 2 + 2 \log _ { 2 } x$$

1. show that $$3 x ^ { 2 } - 13 x - 30 = 0$$
2. 1. Write down the roots of the equation $$3 x ^ { 2 } - 13 x - 30 = 0$$
   2. Hence state which of the roots in part (b)(i) is not a solution of $$\log _ { 2 } ( x + 3 ) + \log _ { 2 } ( x + 10 ) = 2 + 2 \log _ { 2 } x$$ giving a reason for your answer.

1. The function \(g\) is defined by

$$g ( x ) = \frac { 3 \ln ( x ) - 7 } { \ln ( x ) - 2 } \quad x > 0 \quad x \neq k$$ where \(k\) is a constant.

1. Deduce the value of \(k\).
2. Prove that $$\mathrm { g } ^ { \prime } ( x ) > 0$$ for all values of \(x\) in the domain of g .
3. Find the range of values of \(a\) for which $$g ( a ) > 0$$

1. Using the laws of logarithms, solve the equation

$$\log _ { 3 } ( 12 y + 5 ) - \log _ { 3 } ( 1 - 3 y ) = 2$$

8

1. Show that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\), where \(k\) is a constant, can be expressed in the form \(x ^ { 2 } - 8 k x + 8 = 0\).
2. Given that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\) has only one real root, find the value of this root.

6 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-05_538_531_264_246} The shape \(A B C\) shown in the diagram is a student's design for the sail of a small boat.
The curve \(A C\) has equation \(y = 2 \log _ { 2 } x\) and the curve \(B C\) has equation \(y = \log _ { 2 } \left( x - \frac { 3 } { 2 } \right) + 3\).

1. State the \(x\)-coordinate of point \(A\).
2. Determine the \(x\)-coordinate of point \(B\).
3. By solving an equation involving logarithms, show that the \(x\)-coordinate of point \(C\) is 2 . It is given that, correct to 3 significant figures, the area of the sail is 0.656 units \(^ { 2 }\).
4. Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines.

11 The intensity of the sun's radiation, \(y\) watts per square metre, and the average distance from the sun, \(x\) astronomical units, are shown in Fig. 11 for the planets Mercury and Jupiter. \begin{table}[h] \end{table} The intensity \(y\) is proportional to a power of the distance \(x\).

1. Write down an equation for \(y\) in terms of \(x\) and two constants.
2. Show that the equation can be written in the form \(\ln y = a + b \ln x\).
3. In the Printed Answer Booklet, complete the table for \(\ln x\) and \(\ln y\) correct to 4 significant figures.
4. Use the values from part (iii) to find \(a\) and \(b\).
5. Hence rewrite your equation from part (i) for \(y\) in terms of \(x\), using suitable numerical values for the constants.
6. Sketch a graph of the equation found in part (v).
7. Earth is 1 astronomical unit from the sun. Find the intensity of the sun's radiation for Earth.

5 In 2019 scientists developed a model for comparing the ages of humans and dogs.
According to the model, \(Y = A \ln X + B\) where \(X =\) dog age in years and \(Y =\) human age in years.
For the model, it is known that when \(X = 1 , Y = 31\) and when \(X = 12 , Y = 71\).

1. Find the value of \(B\).
2. Determine the value of \(A\), correct to the nearest whole number. Use the model, with the exact value of \(B\) and the value of \(A\) correct to the nearest whole number, to answer parts (c) and (d).
3. Find the human age corresponding to a dog age of 20 years.
4. Determine the dog age corresponding to a human age of 120 years.

2

1. Express \(2 \log _ { 3 } x + \log _ { 3 } a\) as a single logarithm.
2. Given that \(2 \log _ { 3 } x + \log _ { 3 } a = 2\), express \(x\) in terms of \(a\).

14

1. Use the laws of logarithms to show that \(\log _ { 10 } 200 - \log _ { 10 } 20\) is equal to 1 . The first three terms of a sequence are \(\log _ { 10 } 20 , \log _ { 10 } 200 , \log _ { 10 } 2000\).
2. Show that the sequence is arithmetic.
3. Find the exact value of the sum of the first 50 terms of this sequence.

5

1. Make \(y\) the subject of the formula \(\log _ { 10 } ( y - k ) = x \log _ { 10 } 2\), where \(k\) is a positive constant.
2. Sketch the graph of \(y\) against \(x\).