Laws of Logarithms

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

2

1. Use logarithms to solve the equation \(3 ^ { X } = 8\), giving your answer correct to 2 decimal places.
2. It is given that $$\ln z = \ln ( y + 2 ) - 2 \ln y$$ where \(y > 0\). Express \(z\) in terms of \(y\) in a form not involving logarithms.

10. (i) Use the laws of logarithms to solve the equation $$3 \log _ { 8 } 2 + \log _ { 8 } ( 7 - x ) = 2 + \log _ { 8 } x$$ (ii) Using algebra, find, in terms of logarithms, the exact value of \(y\) for which $$3 ^ { 2 y } + 3 ^ { y + 1 } = 10$$

1. Evaluate $$\log_3 27 - \log_3 4.$$ [4]
2. Solve the equation $$4^x - 3(2^{x+1}) = 0.$$ [5]

3

1. Find the value of \(x\) in each of the following:
   1. \(\quad \log _ { 9 } x = 0\);
   2. \(\quad \log _ { 9 } x = \frac { 1 } { 2 }\).
2. Given that $$2 \log _ { a } n = \log _ { a } 18 + \log _ { a } ( n - 4 )$$ find the possible values of \(n\).

1. Anna, who is confused about the rules for logarithms, states that $$(\log_3 p)^2 = \log_3 (p^2)$$ and $$\log_3(p + q) = \log_3 p + \log_3 q.$$ However, there is a value for \(p\) and a value for \(q\) for which both statements are correct. Find the value of \(p\) and the value of \(q\). [7]
2. Solve $$\frac{\log_3(3x^3 - 23x^2 + 40x)}{\log_3 9} = 0.5 + \log_3(3x - 8).$$ [7]

2

1. Show that the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$ can be written as a quadratic equation in \(x\).
2. Hence solve the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$

2

1. Show that the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$ can be written as a quadratic equation in \(x\).
2. Hence solve the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$

6. Given that $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$

1. show that $$4 x ^ { 2 } - 16 x - 9 = 0$$
2. Hence solve the equation $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$

1 Solve the equation $$\ln ( x + 1 ) - \ln x = 2 \ln 2$$

4

1. Express \(2 \log _ { 3 } x - \log _ { 3 } ( x + 4 )\) as a single logarithm.
2. Hence solve the equation \(2 \log _ { 3 } x - \log _ { 3 } ( x + 4 ) = 2\).

2 Solve the equation \(\ln ( x - 5 ) = 7 - \ln x\). Give your answer correct to 2 decimal places.

2 Solve the equation \(\ln ( 3 - 2 x ) - 2 \ln x = \ln 5\).

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]

2 Solve the equation \(\ln ( 3 - 2 x ) - 2 \ln x = \ln 5\).

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

11.

1. Given that \(\log _ { 3 } c = m\) and \(\log _ { 27 } d = n\), express \(\frac { \sqrt { c } } { d ^ { 2 } }\) in the form \(3 ^ { y }\), where \(y\) is an expression in terms of \(m\) and \(n\).
2. Show that the equation $$\log _ { 4 } ( 2 x + 3 ) + \log _ { 4 } ( 2 x + 15 ) = 1 + \log _ { 4 } ( 14 x + 5 )$$ has only one solution and state its value.

1. Use the laws of logarithms to solve

$$\log _ { 2 } ( 16 x ) + \log _ { 2 } ( x + 1 ) = 3 + \log _ { 2 } ( x + 6 )$$

1.(a)Solve the equation $$\sqrt { } ( 3 x + 16 ) = 3 + \sqrt { } ( x + 1 )$$ (b)Solve the equation $$\log _ { 3 } ( x - 7 ) - \frac { 1 } { 2 } \log _ { 3 } x = 1 - \log _ { 3 } 2$$

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

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

6.(a)Show that $$\sqrt { 2 + \sqrt { 3 } } - \sqrt { 2 - \sqrt { 3 } } = \sqrt { 2 }$$ (b)Hence prove that $$\log _ { \frac { 1 } { 8 } } ( \sqrt { 2 + \sqrt { 3 } } - \sqrt { 2 - \sqrt { 3 } } ) = - \frac { 1 } { 6 } .$$ (c)Find all possible pairs of integers \(a\) and \(n\) such that $$\log _ { \frac { 1 } { n } } ( \sqrt { a + \sqrt { 15 } } - \sqrt { a - \sqrt { 15 } } ) = - \frac { 1 } { 2 } .$$

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

3 Express each of the following as a single logarithm:

1. \(\log _ { a } 2 + \log _ { a } 3\),
2. \(2 \log _ { 10 } x - 3 \log _ { 10 } y\).

3

1. Write \(\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 )\) as a single logarithm.
2. Without using your calculator, verify that \(x = 4\) is a root of the equation $$\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 ) = 1$$

$$\begin{gathered} 2 \log _ { 2 } y = 5 - \log _ { 2 } x \\ \log _ { x } y = - 3 \end{gathered}$$ for \(x > 0 , y > 0\)

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 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]

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 \(a\).

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 \(a\).

1. Given that \(y = \log_2 x\), find expressions in terms of \(y\) for
   1. \(\log_2 \left(\frac{x}{2}\right)\), [2]
   2. \(\log_2 (\sqrt{x})\). [2]
2. Hence, or otherwise, solve the equation $$2 \log_2 \left(\frac{x}{2}\right) + \log_2 (\sqrt{x}) = 8.$$ [3]

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

1. Simplify \(\log_a 1 - \log_a (a^m)^3\). [2]
2. Use logarithms to solve the equation \(3^{2x+1} = 1000\). Give your answer correct to 3 significant figures. [3]

Solve the equation \(2^x = 4^{2x+1}\). [3]

Write down the values of \(\log_a a\) and \(\log_a (a^3)\). [2]

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

The points \((2, 6)\) and \((3, 18)\) lie on the curve \(y = ax^n\). Use logarithms to find the values of \(a\) and \(n\), giving your answers correct to 2 decimal places. [5]

A student was asked to solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\). The student's attempt is written out below. \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\) \(4\log_3 x - 3 \log_3 x - 2 = 0\) \(\log_3 x - 2 = 0\) \(\log_3 x = 2\) \(x = 8\)

1. Identify the two mistakes that the student has made. [2]
2. Solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\), giving your answers in an exact form. [4]

5. (a) Given that \(t = \log _ { 3 } x\), find expressions in terms of \(t\) for

1. \(\log _ { 3 } x ^ { 2 }\),
2. \(\log _ { 9 } x\).
   (b) Hence, or otherwise, find to 3 significant figures the value of \(x\) such that $$\log _ { 3 } x ^ { 2 } - \log _ { 9 } x = 4 .$$

Make \(x\) the subject of \(t = \ln \sqrt{\frac{5}{(x-3)}}\). [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]

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

The graph of \(\log_{10} y\) against \(x\) is a straight line with gradient 2 and the intercept on the vertical axis at 4. Write down an equation for this straight line and show that \(y = 10000 \times 100^x\). [4]

Solve the equation $$2\ln(5 - e^{-2x}) = 1,$$ giving your answer correct to 3 significant figures. [4]

5

1. (A) Sketch the graph of \(y = 3 ^ { x }\).
   (B) Give the coordinates of any intercepts. The curve \(y = \mathrm { f } ( x )\) is the reflection of the curve \(y = 3 ^ { x }\) in the line \(y = x\).
2. Find \(\mathrm { f } ( x )\).

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

A function \(f\) is defined by \(f(x) = \log_{10}(2 - x)\). Another function \(g\) is defined by \(g(x) = \log_{10}(5 - x)\). The diagram below shows a sketch of the graphs of \(y = f(x)\) and \(y = g(x)\). \includegraphics{figure_17}

1. The point \((c, 1)\) lies on \(y = f(x)\). Find the value of \(c\). [2]
2. A point P lies on \(y = f(x)\) and has \(x\)-coordinate \(\alpha\). Another point Q lies on \(y = g(x)\) and also has \(x\)-coordinate \(\alpha\). The distance between P and Q is 1.2 units. Find the value of \(\alpha\), giving your answer correct to three decimal places. [5]

1. The weight of a baby mammal is monitored over a 16 -month period.

The weight of the mammal, \(w \mathrm {~kg}\), is given by $$w = \log _ { a } ( t + 5 ) - \log _ { a } 4 \quad 2 \leqslant t \leqslant 18$$ where \(t\) is the age of the mammal in months and \(a\) is a constant.
Given that the weight of the mammal was 10 kg when \(t = 3\)

1. show that \(a = 1.072\) correct to 3 decimal places. Using \(a = 1.072\)
2. find an equation for \(t\) in terms of \(w\)
3. find the value of \(t\) when \(w = 15\), giving your answer to 3 significant figures.

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.

The points \((2, 6)\) and \((3, 18)\) lie on the curve \(y = ax^n\). Use logarithms to find the values of \(a\) and \(n\), giving your answers correct to 2 decimal places. [5]

The graph of \(\log_{10} y\) against \(x\) is a straight line with gradient 2 and the intercept on the vertical axis at 4. Write down an equation for this straight line and show that \(y = 10000 \times 100^x\). [4]