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

2. Find the exact solutions, in their simplest form, to the equations

1. \(\mathrm { e } ^ { 3 x - 9 } = 8\)
2. \(\ln ( 2 y + 5 ) = 2 + \ln ( 4 - y )\) \includegraphics[max width=\textwidth, alt={}, center]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-05_37_1813_0_6}

3. Find the exact solutions of

1. \(\mathrm { e } ^ { 2 x + 3 } = 6\),
2. \(\ln ( 3 x + 2 ) = 4\).

Solve

1. \(5 ^ { x } = 8\), giving your answer to 3 significant figures,
2. \(\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7\).

5 Solve the following equations.

1. \(\quad 2 ^ { x } = \frac { 1 } { 8 }\).
2. \(\quad x ^ { - \frac { 1 } { 2 } } = \frac { 1 } { 4 }\)

1. Find the value of \(y\) such that

$$4 ^ { y + 3 } = 8$$

1. Solve the equation

$$9 ^ { x } = 3 ^ { x + 2 } .$$

3 The profit \(\pounds P\) made by a company in its \(n\)th year is modelled by the exponential function $$P = A \mathrm { e } ^ { b n }$$ In the first year (when \(n = 1\) ), the profit was \(\pounds 10000\). In the second year, the profit was \(\pounds 16000\).

1. Show that \(\mathrm { e } ^ { b } = 1.6\), and find \(b\) and \(A\).
2. What does this model predict the profit to be in the 20th year?

4 The temperature \(T ^ { \circ } \mathrm { C }\) of a liquid at time \(t\) minutes is given by the equation $$T = 30 + 20 \mathrm { e } ^ { - 0.05 t } , \quad \text { for } t \geqslant 0 .$$ Write down the initial temperature of the liquid, and find the initial rate of change of temperature.
Find the time at which the temperature is \(40 ^ { \circ } \mathrm { C }\).

8

1. On a single diagram, sketch the curves with the following equations. In each case state the coordinates of any points of intersection with the axes.
   1. \(y = a ^ { x }\), where \(a\) is a constant such that \(a > 1\).
   2. \(y = 2 b ^ { x }\), where \(b\) is a constant such that \(0 < b < 1\).
   3. The curves in part (i) intersect at the point \(P\). Prove that the \(x\)-coordinate of \(P\) is $$\frac { 1 } { \log _ { 2 } a - \log _ { 2 } b } .$$

5 In a geometric progression, the first term is 5 and the second term is 4.8 .

1. Show that the sum to infinity is 125 .
2. The sum of the first \(n\) terms is greater than 124 . Show that $$0.96 ^ { n } < 0.008$$ and use logarithms to calculate the smallest possible value of \(n\).

8 The first term of a geometric progression is 10 and the common ratio is 0.8.

1. Find the fourth term.
2. Find the sum of the first 20 terms, giving your answer correct to 3 significant figures.
3. The sum of the first \(N\) terms is denoted by \(S _ { N }\), and the sum to infinity is denoted by \(S _ { \infty }\). Show that the inequality \(S _ { \infty } - S _ { N } < 0.01\) can be written as $$0.8 ^ { N } < 0.0002 ,$$ and use logarithms to find the smallest possible value of \(N\).

3 U se logarithms to solve the equation \(3 ^ { 2 x + 1 } = 5 ^ { 200 }\), giving the value of \(x\) correct to 3 significant figures.

8

1. Solve the equation \(10 ^ { x } = 316\).
2. Simplify \(\log _ { a } \left( a ^ { 2 } \right) - 4 \log _ { a } \left( \frac { 1 } { a } \right)\).

6 Sketch the graph of \(y = 2 ^ { x }\).
Solve the equation \(2 ^ { x } = 50\), giving your answer correct to 2 decimal places.

11 There is a flowerhead at the end of each stem of an oleander plant. The next year, each flowerhead is replaced by three stems and flowerheads, as shown in Fig. 11. \begin{figure}[h] \end{figure}

1. How many flowerheads are there in year 5?
2. How many flowerheads are there in year \(n\) ?
3. As shown in Fig. 11, the total number of stems in year 2 is 4, (that is, 1 old one and 3 new ones). Similarly, the total number of stems in year 3 is 13 , (that is, \(1 + 3 + 9\) ). Show that the total number of stems in year \(n\) is given by \(\frac { 3 ^ { n } - 1 } { 2 }\).
4. Kitty's oleander has a total of 364 stems. Find
   (A) its age,
   (B) how many flowerheads it has.
5. Abdul's oleander has over 900 flowerheads. Show that its age, \(y\) years, satisfies the inequality \(y > \frac { \log _ { 10 } 900 } { \log _ { 10 } 3 } + 1\).
   Find the smallest integer value of \(y\) for which this is true.

7

1. Sketch the graph of \(y = 3 ^ { x }\).
2. Use logarithms to solve the equation \(3 ^ { x } = 20\). Give your answer correct to 2 decimal places.

2

1. Write \(\log _ { 2 } 5 + \log _ { 2 } 1.6\) as an integer.
2. Solve the equation \(2 ^ { x } = 3\), giving your answer correct to 4 decimal places.

2 Solve the equation \(3 ^ { x } = 15\), giving your answer correct to 4 decimal places.

6 Find the solution to this equation, correct to 3 significant figures. $$\left( 2 ^ { x } \right) \left( 2 ^ { x + 1 } \right) = 10 .$$

5.

1. Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$
2. Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0$$

4.

1. Solve the inequality $$x ^ { 2 } - 13 x + 30 < 0$$
2. Hence find the set of values of \(y\) such that $$2 ^ { 2 y } - 13 \left( 2 ^ { y } \right) + 30 < 0 .$$

1. Solve the equation

$$\log _ { 5 } ( 4 x + 3 ) - \log _ { 5 } ( x - 1 ) = 2$$

5.

1. Find the value of \(a\) such that $$\log _ { a } 27 = 3 + \log _ { a } 8$$
2. Solve the equation $$2 ^ { x + 3 } = 6 ^ { x - 1 }$$ giving your answer to 3 significant figures.

4. Giving your answers to 2 decimal places, solve the simultaneous equations $$\begin{aligned} & \mathrm { e } ^ { 2 y } - x + 2 = 0 \\ & \ln ( x + 3 ) - 2 y - 1 = 0 \end{aligned}$$