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

1 Solve the equation \(\ln ( 2 x - 1 ) = 2 \ln ( x + 1 ) - \ln x\). Give your answer correct to 3 decimal places.

1 Find the set of values of \(x\) satisfying the inequality \(\left| 2 ^ { x + 1 } - 2 \right| < 0.5\), giving your answer to 3 significant figures.

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. By substituting \(p = 3 ^ { x }\), show that the equation $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$ can be rewritten in the form $$9 p ^ { 2 } + 26 p - 3 = 0$$
2. Hence solve $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. By substituting \(p = 2 ^ { x }\), show that the equation $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$ can be written in the form $$4 p ^ { 2 } - 33 p + 8 = 0$$
2. Hence solve $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$

9. The resident population of a city is 130000 at the end of Year 1 A model predicts that the resident population of the city will increase by \(2 \%\) each year, with the populations at the end of each year forming a geometric sequence.

1. Show that the predicted resident population at the end of Year 2 is 132600
2. Write down the value of the common ratio of the geometric sequence. The model predicts that Year \(N\) will be the first year which will end with the resident population of the city exceeding 260000
3. Show that $$N > \frac { \log _ { 10 } 2 } { \log _ { 10 } 1.02 } + 1$$
4. Find the value of \(N\).
   

8. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } - 23 x - 10$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ).
2. Show that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).
3. Hence fully factorise \(\mathrm { f } ( x )\).
4. Hence solve $$2 \left( 3 ^ { 3 t } \right) - 5 \left( 3 ^ { 2 t } \right) - 23 \left( 3 ^ { t } \right) = 10$$ giving your answer to 3 decimal places.

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 $$3 ^ { 2 y } + 3 ^ { y + 1 } = 10$$

3. Solve, giving each answer to 3 significant figures, the equations

1. \(4 ^ { a } = 20\)
2. \(3 + 2 \log _ { 2 } b = \log _ { 2 } ( 30 b )\) (Solutions based entirely on graphical or numerical methods are not acceptable.)

1. (i) Find, giving your answer to 3 significant figures, the value of \(y\) for which

$$3 ^ { y } = 12$$ (ii) Solve, giving an exact answer, the equation $$\log _ { 2 } ( x + 3 ) - \log _ { 2 } ( 2 x + 4 ) = 4$$ (You should show each step in your working.)

6. Find the exact values of \(x\) for which $$2 \log _ { 5 } ( x + 5 ) - \log _ { 5 } ( 2 x + 2 ) = 2$$ Give your answers as simplified surds.

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

1. \(7 ^ { 2 x } = 14\)
2. \(\log _ { 5 } ( 3 x + 1 ) = - 2\)

13. (i) Find the value of \(x\) for which $$4 ^ { 3 x + 2 } = 3 ^ { 600 }$$ giving your answer to 4 significant figures.
(ii) Given that $$\log _ { a } ( 3 b - 2 ) - 2 \log _ { a } 5 = 4 , \quad a > 0 , a \neq 1 , b > \frac { 2 } { 3 }$$ find an expression for \(b\) in terms of \(a\).

9. Given that \(y = 3 x ^ { 2 }\),

1. show that \(\log _ { 3 } y = 1 + 2 \log _ { 3 } x\)
2. Hence, or otherwise, solve the equation $$1 + 2 \log _ { 3 } x = \log _ { 3 } ( 28 x - 9 )$$

6. (a) Given \(y = 2 ^ { x }\), show that $$2 ^ { 2 x + 1 } - 17 \left( 2 ^ { x } \right) + 8 = 0$$ can be written in the form $$2 y ^ { 2 } - 17 y + 8 = 0$$ (b) Hence solve $$2 ^ { 2 x + 1 } - 17 \left( 2 ^ { x } \right) + 8 = 0$$

8. (i) Find the exact solution of the equation $$8 ^ { 2 x + 1 } = 6$$ giving your answer in the form \(a + b \log _ { 2 } 3\), where \(a\) and \(b\) are constants to be found.
(ii) Using the laws of logarithms, solve $$\log _ { 5 } ( 7 - 2 y ) = 2 \log _ { 5 } ( y + 1 ) - 1$$

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = 4 ^ { x }\) A copy of Figure 1, labelled Diagram 1, is shown on the next page.

1. On Diagram 1, sketch the curve with equation
   1. \(y = 2 ^ { x }\)
   2. \(y = 4 ^ { x } - 6\) Label clearly the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 2 ^ { x }\) meets the curve with equation \(y = 4 ^ { x } - 6\) at the point \(P\).
2. Using algebra, find the exact coordinates of \(P\).
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   \section*{Diagram 1}

9. (i) Find the exact value of \(x\) for which $$\log _ { 3 } ( x + 5 ) - 4 = \log _ { 3 } ( 2 x - 1 )$$ (ii) Given that $$3 ^ { y + 3 } \times 2 ^ { 1 - 2 y } = 108$$

1. show that $$0.75 ^ { y } = 2$$
2. Hence find the value of \(y\), giving your answer to 3 decimal places.

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7. (i) A geometric sequence has first term 4 and common ratio 6 Given that the \(n ^ { \text {th } }\) term is greater than \(10 ^ { 100 }\), find the minimum possible value of \(n\).
(ii) A different geometric sequence has first term \(a\) and common ratio \(r\). Given that

• the second term of the sequence is - 6
• the sum to infinity of the series is 25
  1. show that

$$25 r ^ { 2 } - 25 r - 6 = 0$$

Write down the solutions of $$25 r ^ { 2 } - 25 r - 6 = 0$$ Hence,

state the value of \(r\), giving a reason for your answer,

find the sum of the first 4 terms of the series. \includegraphics[max width=\textwidth, alt={}, center]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-23_70_37_2617_1914}

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.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Solve $$3 ^ { a } = 70$$ giving the answer to 3 decimal places.
2. Find the exact value of \(b\) such that $$4 + 3 \log _ { 3 } b = \log _ { 3 } 5 b$$

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

1. \(3 ^ { x } = 5\),
2. \(\log _ { 2 } ( 2 x + 1 ) - \log _ { 2 } x = 2\).

4. Solve the equation $$5 ^ { x } = 17$$ giving your answer to 3 significant figures.