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

3. Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which

1. \(3 ^ { x } = 5\),
2. \(\log _ { 2 } ( 2 x + 1 ) - \log _ { 2 } x = 2\).

4. Given that \(0 < x < 4\) and $$\log _ { 5 } ( 4 - x ) - 2 \log _ { 5 } x = 1$$ find the value of \(x\).
(6)

5. (a) Find the positive value of \(x\) such that $$\log _ { x } 64 = 2$$ (b) Solve for \(x\) $$\log _ { 2 } ( 11 - 6 x ) = 2 \log _ { 2 } ( x - 1 ) + 3$$

4. Given that \(y = 3 x ^ { 2 }\),

1. show that \(\log _ { 3 } y = 1 + 2 \log _ { 3 } x\)
2. Hence, or otherwise, solve the equation $$1 + 2 \log _ { 3 } x = \log _ { 3 } ( 28 x - 9 )$$

6. Given that $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$

1. Show that $$x ^ { 2 } - 34 x + 225 = 0$$
2. Hence, or otherwise, solve the equation $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$

6. Given that $$\log _ { x } ( 7 y + 1 ) - \log _ { x } ( 2 y ) = 1 , \quad x > 4 , \quad 0 < y < 1$$ express \(y\) in terms of \(x\).

6. (a) Find, to 3 significant figures, the value of \(x\) for which \(8 ^ { x } = 0.8\).
(b) Solve the equation $$2 \log _ { 3 } x - \log _ { 3 } 7 x = 1$$

8. A trading company made a profit of \(\pounds 50000\) in 2006 (Year 1). A model for future trading predicts that profits will increase year by year in a geometric sequence with common ratio \(r , r > 1\). The model therefore predicts that in 2007 (Year 2) a profit of \(\pounds 50000 r\) will be made.

1. Write down an expression for the predicted profit in Year \(n\). The model predicts that in Year \(n\), the profit made will exceed \(\pounds 200000\).
2. Show that \(n > \frac { \log 4 } { \log r } + 1\). Using the model with \(r = 1.09\),
3. find the year in which the profit made will first exceed \(\pounds 200000\),
4. find the total of the profits that will be made by the company over the 10 years from 2006 to 2015 inclusive, giving your answer to the nearest \(\pounds 10000\).

4. (a) Find, to 3 significant figures, the value of \(x\) for which \(5 ^ { x } = 7\).
(b) Solve the equation \(5 ^ { 2 x } - 12 \left( 5 ^ { x } \right) + 35 = 0\).

8. (a) Find the value of \(y\) such that $$\log _ { 2 } y = - 3$$ (b) Find the values of \(x\) such that $$\frac { \log _ { 2 } 32 + \log _ { 2 } 16 } { \log _ { 2 } x } = \log _ { 2 } x$$

7. (a) Given that $$2 \log _ { 3 } ( x - 5 ) - \log _ { 3 } ( 2 x - 13 ) = 1$$ show that \(x ^ { 2 } - 16 x + 64 = 0\).
(b) Hence, or otherwise, solve \(2 \log _ { 3 } ( x - 5 ) - \log _ { 3 } ( 2 x - 13 ) = 1\).

2. Find the values of \(x\) such that $$2 \log _ { 3 } x - \log _ { 3 } ( x - 2 ) = 2$$

6. Given that \(\log _ { 3 } x = a\), find in terms of \(a\),

1. \(\log _ { 3 } ( 9 x )\)
2. \(\log _ { 3 } \left( \frac { x ^ { 5 } } { 81 } \right)\) giving each answer in its simplest form.
3. Solve, for \(x\), $$\log _ { 3 } ( 9 x ) + \log _ { 3 } \left( \frac { x ^ { 5 } } { 81 } \right) = 3$$ giving your answer to 4 significant figures. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4f4eac7b-8908-480f-bb39-049944203fff-10_775_1605_221_159} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} The line with equation \(y = 10\) cuts the curve with equation \(y = x ^ { 2 } + 2 x + 2\) at the points \(A\) and \(B\) as shown in Figure 1. The figure is not drawn to scale.

7. (i) Find the exact value of \(x\) for which $$\log _ { 2 } ( 2 x ) = \log _ { 2 } ( 5 x + 4 ) - 3$$ (ii) Given that $$\log _ { a } y + 3 \log _ { a } 2 = 5$$ express \(y\) in terms of \(a\).
Give your answer in its simplest form.

8. (i) Solve $$5 ^ { y } = 8$$ giving your answer to 3 significant figures.
(ii) Use algebra to find the values of \(x\) for which $$\log _ { 2 } ( x + 15 ) - 4 = \frac { 1 } { 2 } \log _ { 2 } x$$

7. (i) Use logarithms to solve the equation \(8 ^ { 2 x + 1 } = 24\), giving your answer to 3 decimal places.
(ii) Find the values of \(y\) such that $$\log _ { 2 } ( 11 y - 3 ) - \log _ { 2 } 3 - 2 \log _ { 2 } y = 1 , \quad y > \frac { 3 } { 11 }$$

8. (i) Given that $$\log _ { 3 } ( 3 b + 1 ) - \log _ { 3 } ( a - 2 ) = - 1 , \quad a > 2$$ express \(b\) in terms of \(a\).
(ii) Solve the equation $$2 ^ { 2 x + 5 } - 7 \left( 2 ^ { x } \right) = 0$$ giving your answer to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

7. (i) \(2 \log ( x + a ) = \log \left( 16 a ^ { 6 } \right)\), where \(a\) is a positive constant Find \(x\) in terms of \(a\), giving your answer in its simplest form.
(ii) \(\quad \log _ { 3 } ( 9 y + b ) - \log _ { 3 } ( 2 y - b ) = 2\), where \(b\) is a positive constant Find \(y\) in terms of \(b\), giving your answer in its simplest form.

7. (i) Find the value of \(y\) for which $$1.01 ^ { y - 1 } = 500$$ Give your answer to 2 decimal places.
(ii) Given that $$2 \log _ { 4 } ( 3 x + 5 ) = \log _ { 4 } ( 3 x + 8 ) + 1 , \quad x > - \frac { 5 } { 3 }$$

1. show that $$9 x ^ { 2 } + 18 x - 7 = 0$$
2. Hence solve the equation $$2 \log _ { 4 } ( 3 x + 5 ) = \log _ { 4 } ( 3 x + 8 ) + 1 , \quad x > - \frac { 5 } { 3 }$$ DO NOTI WRITE IN THIS AREA

7. (a) Use the factor theorem to show that \(( x + 1 )\) is a factor of \(x ^ { 3 } - x ^ { 2 } - 10 x - 8\).
(b) Find all the solutions of the equation \(x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0\).
(c) Prove that the value of \(x\) that satisfies $$2 \log _ { 2 } x + \log _ { 2 } ( x - 1 ) = 1 + \log _ { 2 } ( 5 x + 4 )$$ is a solution of the equation $$x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0$$ (d) State, with a reason, the value of \(x\) that satisfies equation (I).

2. The function \(f\) and the function \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 12 } { x + 1 } & x > 0 , x \in \mathbb { R } \\ \mathrm {~g} ( x ) = \frac { 5 } { 2 } \ln x & x > 0 , x \in \mathbb { R } \end{array}$$

1. Find, in simplest form, the value of \(\mathrm { fg } \left( \mathrm { e } ^ { 2 } \right)\)
2. Find f-1
3. Hence, or otherwise, find all real solutions of the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )$$

8. A dose of antibiotics is given to a patient. The amount of the antibiotic, \(x\) milligrams, in the patient's bloodstream \(t\) hours after the dose was given, is found to satisfy the equation $$\log _ { 10 } x = 2.74 - 0.079 t$$

1. Show that this equation can be written in the form $$x = p q ^ { - t }$$ where \(p\) and \(q\) are constants to be found. Give the value of \(p\) to the nearest whole number and the value of \(q\) to 2 significant figures.
2. With reference to the equation in part (a), interpret the value of the constant \(p\). When a different dose of the antibiotic is given to another patient, the values of \(x\) and \(t\) satisfy the equation $$x = 400 \times 1.4 ^ { - t }$$
3. Use calculus to find, to 2 significant figures, the value of \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when \(t = 5\)

2. \begin{figure}[h] \end{figure} Figure 1 shows the linear relationship between \(\log _ { 6 } T\) and \(\log _ { 6 } x\) The line passes through the points \(( 0,4 )\) and \(( 2,0 )\) as shown.

1. 1. Find an equation linking \(\log _ { 6 } T\) and \(\log _ { 6 } x\)
   2. Hence find the exact value of \(T\) when \(x = 216\)
2. Find an equation, not involving logs, linking \(T\) with \(x\)

1. (i) The variables \(x\) and \(y\) are connected by the equation

$$y = \frac { 10 ^ { 6 } } { x ^ { 3 } } \quad x > 0$$ Sketch the graph of \(\log _ { 10 } y\) against \(\log _ { 10 } x\) Show on your sketch the coordinates of the points of intersection of the graph with the axes.
(ii) \begin{figure}[h] \end{figure} Figure 2 shows the linear relationship between \(\log _ { 3 } N\) and \(t\).
Show that \(N = a b ^ { t }\) where \(a\) and \(b\) are constants to be found.

1. A scientist monitored the growth of bacteria on a dish over a 30 -day period.

The area, \(N \mathrm {~mm} ^ { 2 }\), of the dish covered by bacteria, \(t\) days after monitoring began, is modelled by the equation $$\log _ { 10 } N = 0.0646 t + 1.478 \quad 0 \leqslant t \leqslant 30$$

1. Show that this equation may be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(a\) to the nearest integer and give the value of \(b\) to 3 significant figures.
2. Use the model to find the area of the dish covered by bacteria 30 days after monitoring began. Give your answer, in \(\mathrm { mm } ^ { 2 }\), to 2 significant figures.