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

Let \(\text{f}(x) = \frac{4x^2 + 9x - 8}{(x + 2)(2x - 1)}\).

1. Express \(\text{f}(x)\) in the form \(A + \frac{B}{x + 2} + \frac{C}{2x - 1}\). [4]
2. Hence show that \(\int_1^4 \text{f}(x) \, dx = 6 + \frac{1}{2} \ln\left(\frac{16}{7}\right)\). [5]

In this question you must show all stages of your working. Give your answers in fully simplified surd form. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations \begin{align} a - b &= 8
\log_5 a + \log_5 b &= 3 \end{align} [6]

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

1. \(3^x = 5\), [3]
2. \(\log_2(2x + 1) - \log_2 x = 2\). [4]

Solve

1. \(5^x = 8\), giving your answer to 3 significant figures, [3]
2. \(\log_2(x + 1) - \log_2 x = \log_2 7\). [3]

A trading company made a profit of £50 000 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 £50 000r will be made.

1. Write down an expression for the predicted profit in Year \(n\). [1]

The model predicts that in Year \(n\), the profit made will exceed £200 000.

1. Show that \(n > \frac{\log 4}{\log r} + 1\). [3]

Using the model with \(r = 1.09\),

1. find the year in which the profit made will first exceed £200 000, [2]
2. 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 £10 000. [3]

Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations \(a = 3b\), \(\log_3 a + \log_3 b = 2\). Give your answers as exact numbers. [6]

Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations $$a = 3b,$$ $$\log_3 a + \log_3 b = 2.$$ Give your answers as exact numbers. [6]

Solve $$2 \log_3 x - \log_3 (x - 2) = 2, \quad x > 2.$$ [6]

Given that \(p = \log_4 16\), express in terms of \(p\),

1. \(\log_4 2\). [2]
2. \(\log_4 (8q)\). [4]

The sequence \(u_1, u_2, u_3, \ldots, u_n\) is defined by the recurrence relation $$u_{n+1} = pu_n + 5, \quad u_1 = 2, \text{ where } p \text{ is a constant.}$$ Given that \(u_3 = 8\),

1. show that one possible value of \(p\) is \(\frac{1}{2}\) and find the other value of \(p\). [5]

Using \(p = \frac{1}{2}\),

1. write down the value of \(\log_2 p\). [1]

Given also that \(\log_2 q = t\),

1. express \(\log_2 \left(\frac{p^3}{\sqrt{q}}\right)\) in terms of \(t\). [3]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = e^{-x} - 1\).

1. Copy Fig. 1 and on the same axes sketch the graph of \(y = \frac{1}{2}|x - 1|\). Show the coordinates of the points where the graph meets the axes. [2]

The \(x\)-coordinate of the point of intersection of the graphs is \(\alpha\).

1. Show that \(x = \alpha\) is a root of the equation \(x + 2e^{-x} - 3 = 0\). [3]
2. Show that \(-1 < \alpha < 0\). [2]

The iterative formula \(x_{n+1} = -\ln[\frac{1}{2}(3 - x_n)]\) is used to solve the equation \(x + 2e^{-x} - 3 = 0\).

1. Starting with \(x_0 = -1\), find the values of \(x_1\) and \(x_2\). [2]
2. Show that, to 2 decimal places, \(\alpha = -0.58\). [2]

Given that \(p = \log_q 16\), express in terms of \(p\),

1. \(\log_q 2\). [2]
2. \(\log_q (8q)\). [4]

Given that log₂ x = a, find, in terms of a, the simplest form of

1. log₂ (16x), [2]
2. log₂ \(\left(\frac{x⁴}{2}\right)\). [3]

1. Hence, or otherwise, solve $$\log_2 (16x) - \log_2 \left(\frac{x^4}{2}\right) = \frac{1}{2},$$ giving your answer in its simplest surd form. [4]

The sequence \(u_1, u_2, u_3, \ldots, u_n\) is defined by the recurrence relation $$u_{n+1} = pu_n + 5, u_1 = 2, \text{ where } p \text{ is a constant.}$$ Given that \(u_3 = 8\),

1. show that one possible value of \(p\) is \(\frac{1}{2}\) and find the other value of \(p\). [5]

Using \(p = \frac{1}{2}\),

1. write down the value of \(\log_2 p\). [1]

Given also that \(\log_2 q = t\),

1. express \(\log_2 \left(\frac{p^3}{\sqrt{q}}\right)\) in terms of \(t\). [3]

Given that \(\log_2 x = a\), find, in terms of \(a\), the simplest form of

1. \(\log_2 (16x)\), [2]
2. \(\log_2 \left( \frac{x^4}{2} \right)\). [3]
3. Hence, or otherwise, solve \(\log_2 (16x) - \log_2 \left( \frac{x^4}{2} \right) = \frac{1}{2}\), giving your answer in its simplest surd form. [4]

Express \(\log_a x^3 + \log_a \sqrt{x}\) in the form \(k \log_a x\). [2]

\includegraphics{figure_12} A branching plant has stems, nodes, leaves and buds. • There are 7 leaves at each node. • From each node, 2 new stems grow. • At the end of each final stem, there is a bud. Fig. 12 shows one such plant with 3 stages of nodes. It has 15 stems, 7 nodes, 49 leaves and 8 buds.

1. One of these plants has 10 stages of nodes.
   1. How many buds does it have? [2]
   2. How many stems does it have? [2]
2. 1. Show that the number of leaves on one of these plants with \(n\) stages of nodes is $$7(2^n - 1).$$ [2]
   2. One of these plants has \(n\) stages of nodes and more than 200000 leaves. Show that \(n\) satisfies the inequality \(n > \frac{\log_{10} 200007 - \log_{10} 7}{\log_{10} 2}\). Hence find the least possible value of \(n\). [4]