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

A student conducts an experiment and records the following data for two variables, \(x\) and \(y\).

\(x\)	1	2	3	4	5	6
\(y\)	14	45	130	1100	1300	3400
\(\log_{10} y\)

The student is told that the relationship between \(x\) and \(y\) can be modelled by an equation of the form \(y = kb^x\)

1. Plot values of \(\log_{10} y\) against \(x\) on the grid below. [2 marks] \includegraphics{figure_10}
2. State, with a reason, which value of \(y\) is likely to have been recorded incorrectly. [1 mark]
3. By drawing an appropriate straight line, find the values of \(k\) and \(b\). [4 marks]

Express as a single logarithm \(2\log_a 6 - \log_a 3\) [2 marks]

Simplify \(\log_a 8^a\) Circle your answer. [1 mark] \(a^3\) \qquad \(2a\) \qquad \(3a\) \qquad \(8a\)

Find the exact solution of the equation \(\ln(x + 1) + \ln(x - 1) = \ln 15 - 2\ln 7\) Fully justify your answer. [5 marks]

It is given that $$3 \log_a x = \log_a 72 - 2 \log_a 3$$ Solve the equation to find the value of \(x\) Fully justify your answer. [4 marks]

Find the value of \(\log_a(a^5) + \log_a\left(\frac{1}{a}\right)\) [2 marks]

Given that \(a > 0\), determine which of these expressions is not equivalent to the others. Circle your answer. [1 mark] $$-2\log_{10}\left(\frac{1}{a}\right) \quad 2\log_{10}(a) \quad \log_{10}(a^2) \quad -4\log_{10}(\sqrt{a})$$

Use logarithms to solve the equation $$5^{x-2} = 7^{1570}$$ Give your answer to two decimal places. [3 marks]

A zookeeper models the median mass of infant monkeys born at their zoo, up to the age of 2 years, by the formula $$y = a + b \log_{10} x$$ where \(y\) is the median mass in kilograms, \(x\) is age in months and \(a\) and \(b\) are constants. The zookeeper uses the data shown below to determine the values of \(a\) and \(b\).

Age in months (\(x\))	3	24
Median mass (\(y\))	6.4	12

1. The zookeeper uses the data for monkeys aged 3 months to write the correct equation $$6.4 = a + b \log_{10} 3$$
   1. Use the data for monkeys aged 24 months to write a second equation. [1 mark]
   2. Show that $$b = \frac{5.6}{\log_{10} 8}$$ [3 marks]
   3. Find the value of \(a\). Give your answer to two decimal places. [1 mark]
2. Use a suitable value for \(x\) to determine whether the model can be used to predict the median mass of monkeys less than one week old. [2 marks]

The sum to infinity of a geometric series is 96 The first term of the series is less than 30 The second term of the series is 18

1. Find the first term and common ratio of the series. [5 marks]
2. 1. Show that the \(n\)th term of the series, \(u_n\), can be written as $$u_n = \frac{3^n}{2^{2n-5}}$$ [4 marks]
   2. Hence show that $$\log_3 u_n = n(1 - 2\log_3 2) + 5\log_3 2$$ [3 marks]

A planet takes \(T\) days to complete one orbit of the Sun. \(T\) is known to be related to the planet's average distance \(d\), in millions of kilometres, from the Sun. A graph of \(\log_{10} T\) against \(\log_{10} d\) is shown with data for Mercury and Uranus labelled. \includegraphics{figure_7}

1. 1. Find the equation of the straight line in the form $$\log_{10} T = a + b \log_{10} d$$ where \(a\) and \(b\) are constants to be found. [3 marks]
   2. Show that $$T = K d^n$$ where K and n are constants to be found. [2 marks]
2. Neptune takes approximately 60 000 days to complete one orbit of the Sun. Use your answer to 7(a)(ii) to find an estimate for the average distance of Neptune from the Sun. [2 marks]

A student is asked to solve the equation $$\log_3 x - \log_3 \sqrt{x - 2} = 1$$ The student's attempt is shown $$\log_3 x - \log_3 \sqrt{x - 2} = 1$$ $$x - \sqrt{x - 2} = 3^1$$ $$x - 3 = \sqrt{x - 2}$$ $$(x - 3)^2 = x - 2$$ $$x^2 - 7x + 11 = 0$$ $$x = \frac{7 + \sqrt{5}}{2} \text{ or } x = \frac{7 - \sqrt{5}}{2}$$

1. Identify the error made by this student, giving a brief explanation. [1]
2. Write out the correct solution. [3]

Write down the value of (A) \(\log_a (a^4)\), [1] (B) \(\log_a \left(\frac{1}{a}\right)\). [1]

Solve the simultaneous equations $$3\log_u(x^2y) - \log_u(x^2y^2) + \log_u\left(\frac{9}{x^2y^2}\right) = \log_u 36,$$ $$\log_u y - \log_u(x + 3) = 0.$$ [8]

Prove that $$\log_a a \times \log_a 19 = \log_a 19$$ whatever the value of the positive constant \(a\). [3]

Evaluate the integral $$\int_0^1 \frac{dx}{\sqrt{2x^2 + 4x + 6}}.$$ [6]

Show that $$\log_a(x^{10}) - 2\log_a\left(\frac{x^3}{4}\right) = 4\log_a(2x)$$ [3]

Sketch the curve \(y = 2^{2x+3}\), stating the coordinates of any points of intersection with the axes. [2] The point \(P\) on the curve \(y = 3^{3x+2}\) has \(y\)-coordinate equal to 180. Use logarithms to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. [2] The curves \(y = 2^{2x+3}\) and \(y = 3^{3x+2}\) intersect at the point \(Q\). Show that the \(x\)-coordinate of \(Q\) can be written as $$x = \frac{3\log_3 2 - 2}{3 - 2\log_3 2}.$$ [3]

In the question you must show detailed reasoning Given that \(\log_a x = \frac{\log_n x}{\log_n a}\), show that the sum of the infinite series, where \(n = 0,1,2...\), $$\log_2 e - \log_4 e + \log_{16} e - \cdots + (-1)^n \log_{2^{2^n}} e + \cdots$$ is $$\frac{1}{\ln(2\sqrt{2})}$$ [5] [Total marks: 65]

Let \(a = \log_2 x\), \(b = \log_2 y\) and \(c = \log_2 z\). Express \(\log_2(xy) - \log_2(\frac{z}{y})\) in terms of \(a\), \(b\) and \(c\). [3]

Let \(a = \log_2 x\), \(b = \log_2 y\) and \(c = \log_2 z\). Express \(\log_2(xy) - \log_2(\frac{z^2}{x})\) in terms of \(a\), \(b\) and \(c\). [3]

In this question you must show detailed reasoning. Solve the following simultaneous equations: $$(\log_3 x)^2 + \log_3(y^2) = 5$$ $$\log_3(\sqrt{3xy^{-1}}) = 2$$ [6]