1.06c Logarithm definition: log_a(x) as inverse of a^x

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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.
   [0pt] [4 marks]

3. (a) Given that \(y = \ln x\), find expressions in terms of \(y\) for

1. \(\quad \log _ { 2 } x\),
2. \(\ln \frac { x ^ { 2 } } { \mathrm { e } }\).
   (b) Hence, or otherwise, solve the equation $$\log _ { 2 } x = 4 - \ln \frac { x ^ { 2 } } { \mathrm { e } } ,$$ giving your answer to 2 decimal places.

5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv 6 x - 1 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} ( x ) \equiv \log _ { 2 } ( 3 x + 1 ) , \quad x \in \mathbb { R } , \quad x > - \frac { 1 } { 3 } \end{aligned}$$

1. Evaluate \(\operatorname { gf } ( 1 )\).
2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
3. Find, in terms of natural logarithms, the solution of the equation $$\mathrm { fg } ^ { - 1 } ( x ) = 2$$

5.[In this question the values of \(a , x\) ,and \(n\) are such that \(a\) and \(x\) are positive real numbers,with \(a > 1 , x \neq a , x \neq 1\) and \(n\) is an integer with \(n > 1\) ] Sam was confused about the rules of logarithms and thought that $$\log _ { a } x ^ { n } = \left( \log _ { a } x \right) ^ { n }$$

1. Given that \(x\) satisfies statement(1)find \(x\) in terms of \(a\) and \(n\) . Sam also thought that $$\log _ { a } x + \log _ { a } x ^ { 2 } + \ldots + \log _ { a } x ^ { n } = \log _ { a } x + \left( \log _ { a } x \right) ^ { 2 } + \ldots + \left( \log _ { a } x \right) ^ { n }$$
2. For \(n = 3 , x _ { 1 }\) and \(x _ { 2 } \left( x _ { 1 } > x _ { 2 } \right)\) are the two values of \(x\) that satisfy statement(2).
   1. Find,in terms of \(a\) ,an expression for \(x _ { 1 }\) and an expression for \(x _ { 2 }\) .
   2. Find the exact value of \(\log _ { a } \left( \frac { x _ { 1 } } { x _ { 2 } } \right)\) .
3. Show that if \(\log _ { a } x\) satisfies statement(2)then $$2 \left( \log _ { a } x \right) ^ { n } - n ( n + 1 ) \log _ { a } x + \left( n ^ { 2 } + n - 2 \right) = 0$$

5 A student was asked to solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\). The student's attempt is written out below. $$\begin{aligned} & 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0 \\ & 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x = 2 \\ & x = 8 \end{aligned}$$

1. Identify the two mistakes that the student has made.
2. Solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\), giving your answers in an exact form.

8. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x + \ln 2 , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto \mathrm { e } ^ { 2 x } , & x \in \mathbb { R } . \end{array}$$

1. Prove that the composite function gf is $$\operatorname { gf } : x \mapsto 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
2. Sketch the curve with equation \(y = \operatorname { gf } ( x )\), and show the coordinates of the point where the curve cuts the \(y\)-axis.
3. Write down the range of gf .
4. Find the value of \(x\) for which \(\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3\), giving your answer to 3 significant figures.

9. (i) Find the exact solutions to the equations

1. \(\ln ( 3 x - 7 ) = 5\),
2. \(3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15\).
   (ii) The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R } , \\ \mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , \quad x > 1 . \end{array}$$
   1. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
   2. Find fg and state its range.

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1. It is given that \(n\) satisfies the equation $$\log _ { a } n = \log _ { a } 3 + \log _ { a } ( 2 n - 1 )$$ Find the value of \(n\).
2. Given that \(\log _ { a } x = 3\) and \(\log _ { a } y - 3 \log _ { a } 2 = 4\) :
   1. express \(x\) in terms of \(a\);
   2. express \(x y\) in terms of \(a\).

5 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } - 9\). The graph cuts the coordinate axes at \(( 0 , a )\) and \(( b , 0 )\). \includegraphics[max width=\textwidth, alt={}, center]{908f530c-076d-47b1-90dd-38dbfe44f898-03_826_924_477_541}

1. State the value of \(a\), and show that \(b = \ln 3\).
2. Show that \(y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81\).
3. The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed, giving your answer in the form \(\pi ( p \ln 3 + q )\), where \(p\) and \(q\) are integers.
4. Sketch the curve with equation \(y = \left| \mathrm { e } ^ { 2 x } - 9 \right|\) for \(x \geqslant 0\).

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1. Given that \(3 \mathrm { e } ^ { x } = 4\), find the exact value of \(x\).
2. 1. By substituting \(y = \mathrm { e } ^ { x }\), show that the equation \(3 \mathrm { e } ^ { x } + 20 \mathrm { e } ^ { - x } = 19\) can be written as \(3 y ^ { 2 } - 19 y + 20 = 0\).
   2. Hence solve the equation \(3 \mathrm { e } ^ { x } + 20 \mathrm { e } ^ { - x } = 19\), giving your answers as exact values. (3 marks)

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Using the laws of logarithms, solve $$\log _ { 4 } ( 12 - 2 x ) = 2 + 2 \log _ { 4 } ( x + 1 )$$

1. The number of bacteria on a surface is being monitored.

The number of bacteria, \(N\), on the surface, \(t\) hours after monitoring began is modelled by the equation $$\log _ { 10 } N = 0.35 t + 2$$ Use the equation of the model to answer parts (a) to (c).

1. Find the initial number of bacteria on the surface.
2. Show that the equation of the model can be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(b\) to 2 decimal places.
3. Hence find the rate of growth of bacteria on the surface exactly 5 hours after monitoring began.

1 It is given that \(8 ^ { 4 x } = 4 ^ { 3 x - 6 }\).

1. By expressing each side as a power of 2 , find the value of \(x\).
2. Write down the value of \(\log _ { 4 } | x |\).

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]

\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]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = f(x), x \geq 0\). The curve meets the coordinate axes at the points \((0, c)\) and \((d, 0)\). In separate diagrams sketch the curve with equation

1. \(y = f^{-1}(x)\), [2]
2. \(y = 3f(2x)\). [3]

Indicate clearly on each sketch the coordinates, in terms of \(c\) or \(d\), of any point where the curve meets the coordinate axes. Given that \(f\) is defined by $$f : x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$

1. state
   1. the value of \(c\),
   2. the range of \(f\). [3]
2. Find the value of \(d\), giving your answer to 3 decimal places. [3]

The function \(g\) is defined by $$g : x \to \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$

1. Find \(fg(x)\), giving your answer in its simplest form. [3]

Given that \(y = \log_a x, x > 0\), where \(a\) is a positive constant,

1. 1. express \(x\) in terms of \(a\) and \(y\), [1]
   2. deduce that \(\ln x = y \ln a\). [1]
2. Show that \(\frac{dy}{dx} = \frac{1}{x \ln a}\). [2]

The curve \(C\) has equation \(y = \log_{10} x, x > 0\). The point \(A\) on \(C\) has \(x\)-coordinate \(10\). Using the result in part (b),

1. find an equation for the tangent to \(C\) at \(A\). [4]

The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).

1. Find the exact \(x\)-coordinate of \(B\). [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]

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