1.06g Equations with exponentials: solve a^x = b

1. Find any real values of \(x\) such that

$$2 \log _ { 4 } ( 2 - x ) - \log _ { 4 } ( x + 5 ) = 1$$

2. The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$\begin{aligned} & C _ { 1 } : \quad y = 2 ^ { 3 x + 2 } \\ & C _ { 2 } : \quad y = 4 ^ { - x } \end{aligned}$$ Show that the \(x\)-coordinate of the point where \(C _ { 1 }\) and \(C _ { 2 }\) intersect is \(\frac { - 2 } { 5 }\).

10. The figure 4 shows the curves \(\mathrm { f } ( x ) = A - B e ^ { - 0.5 x }\) and \(\mathrm { g } ( x ) = 26 + e ^ { 0.5 x }\) \begin{figure}[h] \end{figure} Given that \(\mathrm { f } ( x )\) passes through \(( 0,8 )\) and has an horizontal asymptote \(y = 48\) a. Find the values of \(A\) and \(B\) for \(\mathrm { f } ( x )\) (3)
b. State the range of \(\mathrm { g } ( x )\) (1) The curves \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) meet at the points \(C\) and \(D\) c. Find the \(x\)-coordinates of the intersection points \(C\) and \(D\), in the form \(\ln k\), where \(k\) is an integer.

3. Use the laws of logarithms to solve the equation $$2 + \log _ { 2 } ( 2 x + 1 ) = 2 \log _ { 2 } ( 22 - x )$$

8. \begin{figure}[h] \end{figure} The curves with equation \(y = 21 - 2 ^ { x }\) meet the curve with equation \(y = 2 ^ { 2 x + 1 }\) at the point \(A\) as shown in Figure 2. Find the exact coordinates of point \(A\).

9. A cup of tea is cooling down in a room. The temperature of tea, \(\theta ^ { \circ } \mathrm { C }\), at time \(t\) minutes after the tea is made, is modelled by the equation $$\theta = A + 70 e ^ { - 0.025 t }$$ where \(A\) is a positive constant.
Given that the initial temperature of the tea is \(85 ^ { \circ } \mathrm { C }\) a. find the value of \(A\).
b. Find the temperature of the tea 20 minutes after it is made.
c. Find how long it will take the tea to cool down to \(43 ^ { \circ } \mathrm { C }\).
(4)

2. Solve $$4 ^ { x - 3 } = 6$$ giving your answer in the form \(a + b \log _ { 2 } 3\), where \(a\) and \(b\) are constants to be found.

1. The value, \(\pounds V\), of a vintage car \(t\) years after it was first valued on 1 st January 2001, is modelled by the equation

$$V = A p ^ { t } \quad \text { where } A \text { and } p \text { are constants }$$ Given that the value of the car was \(\pounds 32000\) on 1st January 2005 and \(\pounds 50000\) on 1st January 2012

1. 1. find \(p\) to 4 decimal places,
   2. show that \(A\) is approximately 24800
2. With reference to the model, interpret
   1. the value of the constant \(A\),
   2. the value of the constant \(p\). Using the model,
3. find the year during which the value of the car first exceeds \(\pounds 100000\)

1. Given that \(a > b > 0\) and that \(a\) and \(b\) satisfy the equation

$$\log a - \log b = \log ( a - b )$$

1. show that $$a = \frac { b ^ { 2 } } { b - 1 }$$ (3)
2. Write down the full restriction on the value of \(b\), explaining the reason for this restriction.

1. A scientist is studying the number of bees and the number of wasps on an island.

The number of bees, measured in thousands, \(N _ { b }\), is modelled by the equation $$N _ { b } = 45 + 220 \mathrm { e } ^ { 0.05 t }$$ where \(t\) is the number of years from the start of the study.
According to the model,

1. find the number of bees at the start of the study,
2. show that, exactly 10 years after the start of the study, the number of bees was increasing at a rate of approximately 18 thousand per year. The number of wasps, measured in thousands, \(N _ { w }\), is modelled by the equation $$N _ { w } = 10 + 800 \mathrm { e } ^ { - 0.05 t }$$ where \(t\) is the number of years from the start of the study.
   When \(t = T\), according to the models, there are an equal number of bees and wasps.
3. Find the value of \(T\) to 2 decimal places.

1. By taking logarithms of both sides, solve the equation

$$4 ^ { 3 p - 1 } = 5 ^ { 210 }$$ giving the value of \(p\) to one decimal place.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 11 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\), giving your answer as a simplified surd. The line \(l\) has equation \(y = 3 x + k\) where \(k\) is a constant.
      Given that \(l\) is a tangent to \(C\),
2. find the possible values of \(k\), giving your answers as simplified surds.

5. A curve \(C\) has parametric equations $$x = 2 t - 1 , \quad y = 4 t - 7 + \frac { 3 } { t } , \quad t \neq 0$$ Show that the Cartesian equation of the curve \(C\) can be written in the form $$y = \frac { 2 x ^ { 2 } + a x + b } { x + 1 } , \quad x \neq - 1$$ where \(a\) and \(b\) are integers to be found.

1. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 48\), where \(a\) is a constant

Given that \(\mathrm { f } ( - 6 ) = 0\)

1. 1. show that \(a = 4\)
   2. express \(\mathrm { f } ( x )\) as a product of two algebraic factors. Given that \(2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3\)
2. show that \(x ^ { 3 } + 4 x ^ { 2 } - 4 x + 48 = 0\)
3. hence explain why $$2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3$$ has no real roots.

1. Given

$$2 ^ { x } \times 4 ^ { y } = \frac { 1 } { 2 \sqrt { 2 } }$$ express \(y\) as a function of \(x\).

1. (a) Sketch the curve with equation

$$y = 4 ^ { x }$$ stating any points of intersection with the coordinate axes.
(b) Solve $$4 ^ { x } = 100$$ giving your answer to 2 decimal places.

1. A quantity of ethanol was heated until it reached boiling point.

The temperature of the ethanol, \(\theta ^ { \circ } \mathrm { C }\), at time \(t\) seconds after heating began, is modelled by the equation $$\theta = A - B \mathrm { e } ^ { - 0.07 t }$$ where \(A\) and \(B\) are positive constants.
Given that

• the initial temperature of the ethanol was \(18 ^ { \circ } \mathrm { C }\)
• after 10 seconds the temperature of the ethanol was \(44 ^ { \circ } \mathrm { C }\)
  1. find a complete equation for the model, giving the values of \(A\) and \(B\) to 3 significant figures.

Ethanol has a boiling point of approximately \(78 ^ { \circ } \mathrm { C }\)

Use this information to evaluate the model.

1. Using the laws of logarithms, solve the equation

$$\log _ { 3 } ( 12 y + 5 ) - \log _ { 3 } ( 1 - 3 y ) = 2$$

1. (i) Kayden claims that

$$3 ^ { x } \geqslant 2 ^ { x }$$ Determine whether Kayden's claim is always true, sometimes true or never true, justifying your answer.
(ii) Prove that \(\sqrt { 3 }\) is an irrational number.

7

1. Write down an expression for the gradient of the curve \(y = \mathrm { e } ^ { k x }\).
2. The line L is a tangent to the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x }\) at the point where \(x = 2\). Show that L passes through the point \(( 0,0 )\).
3. Find the coordinates of the point of intersection of the curves \(y = 3 \mathrm { e } ^ { x }\) and \(y = 1 - 2 \mathrm { e } ^ { \frac { 1 } { 2 } x }\).

6 In this question you must show detailed reasoning.

1. Solve the inequality \(x ^ { 2 } + x - 6 > 0\), giving your answer in set notation.
2. Solve the equation \(x ^ { 3 } - 7 x ^ { \frac { 3 } { 2 } } - 8 = 0\).
3. Find the exact solution of the equation \(\left( 3 ^ { x } \right) ^ { 2 } = 3 \times 2 ^ { x }\).

6 During some research the size, \(P\), of a population of insects, at time \(t\) months after the start of the research, is modelled by the following formula. \(P = 100 \mathrm { e } ^ { t }\)

1. Use this model to answer the following.
   1. Find the value of \(P\) when \(t = 4\).
   2. Find the value of \(t\) when the population is 9000 .
2. It is suspected that a more appropriate model would be the following formula. \(P = k a ^ { t }\) where \(k\) and \(a\) are constants.
   1. Show that, using this model, the graph of \(\log _ { 10 } P\) against \(t\) would be a straight line. Some observations of \(t\) and \(P\) gave the following results.

      \(t\)	1	2	3	4	5
      \(P\)	100	500	1800	7000	19000
      \(\log _ { 10 } P\)	2.00	2.70	3.26	3.85	4.28

   2. On the grid in the Printed Answer Booklet, draw a line of best fit for the data points \(\left( t , \log _ { 10 } P \right)\) given in the table.
   3. Hence estimate the values of \(k\) and \(a\).

8

1. Show that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\), where \(k\) is a constant, can be expressed in the form \(x ^ { 2 } - 8 k x + 8 = 0\).
2. Given that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\) has only one real root, find the value of this root.

5 \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-4_591_547_262_242} The diagram shows the graphs of \(y = 2 ^ { 3 x }\) and \(y = 2 ^ { 3 x + 2 }\). The graph of \(y = 2 ^ { 3 x }\) can be transformed to the graph of \(y = 2 ^ { 3 x + 2 }\) by means of a stretch.

1. Give details of the stretch. The point \(A\) lies on \(y = 2 ^ { 3 x }\) and the point \(B\) lies on \(y = 2 ^ { 3 x + 2 }\). The line segment \(A B\) is parallel to the \(y\)-axis and the difference between the \(y\)-coordinates of \(A\) and \(B\) is 36 .
2. Determine the \(x\)-coordinate of \(A\). Give your answer in the form \(m \log _ { 2 } n\) where \(m\) and \(n\) are constants to be determined.