1. Sketch the following graphs on the same set of axes:
$y = 3^x$
$y = 0.5^x$
2. Sketch the following graphs on the same set of axes:
$y = \mathrm{e}^{x+1}$
$y = \mathrm{e}^x+1$
3. The graph of $y = \mathrm{e}^{x-1}$ is reflected in the line $y = x$. Write down the equation of the reflection.
$y = 1 + \ln x$
4. The graph of $y = 3\mathrm{e}^{2x+1}$ is reflected in the line $y = x$. Write down the equation of the reflection.
$y = \dfrac{1}{2}(\ln\frac{x}{3}-1)$
5. Solve:
$100 = 20\times5^{2x}$
$3+4^{3x}=7\times4^{3x}$
$\dfrac{1}{2}$
$-\dfrac{1}{6}$
6. Solve:
$2\log x + 3 = 7$
$2\log_{3}(6x)=8$
$100$
$\dfrac{27}{2}$
7. Solve:
$\log(3x-2)-\log2=\log(x+4)$
$2\log(2x) - \log(x+4) = 1$
$10$
$\dfrac{5+\sqrt{185}}{4}$
8. Solve:
$\log_23+\log_2x=2$
$2\log_5x=1+\log_56$
$\dfrac{4}{3}$
$\sqrt{30}$
9. Solve:
$2\log_9(x+1)=2\log_9(2x-3)+1$
$\log_3(x+1)=1+2\log_3(x-1)$
$2$
$2$
10. Given that $a$ and $b$ are positive constants, solve the simultaneous equations: $$a+b=13$$ $$\log_6a+\log_6b=2$$
$a = 4, 9$ and $b = 9, 4$
11. Find the exact solutions to the following:
$5^x=2^{x+1}$
$3^{x+5}=6^x$
$x = \dfrac{1}{\log_25-1}$
$x = \dfrac{5}{\log_36-1}$
12. Given $y = 3\times 5^x$, find the slope and intercept of a graph of $\log y$ against $x$.
Slope $\log 5$ and intercept $\log 3$
13. The value of a car, $V$ is modelled by $$V = 5000\times 2^{-0.1t}$$ where $t$ is the age of the car in years.
Find the value of the car when new.
Find how many years it takes for the car to depreciate to half its original price.
$5000$
$10$
14. The number of people living in a town, $n$ thousands, is modelled by $$n = 5+12\mathrm{e}^{0.05t}$$ where $t$ is the number of years after the year 2025.
What is the initial population?
John wants to estimate the number of people in the town in the year 3025. Suggest one reason why this model might be inappropriate.
$17000$
The model suggests the population will keep growing forever, which is unlikely.
15. The number of people who have a disease at a school, $N$, is modelled by $$N = 1000 - 700e^{-0.1t}$$ where $t$ is the time in months after September. How many people are predicted to have the disease at the start of March?
$575$
16. A student propses two models for the price of a car, $p$ thousands of pounds, over time, $t$: $$p = 30\mathrm{e}^{-0.15t}$$ $$p = 27\mathrm{e}^{-0.13t} + 3$$ Suggest why the second model is more realistic.
It is unlikely that the price of a car will fall to zero.
17. The atmospheric pressure on earth, $p$, can be modelled by $$p = \mathrm{e}^{-0.13h}$$ where $h$ is the height above sea level in kilometres.
Find the pressure at the top of Mount Everest, which has a height of 8849 metres.
How tall, in metres, would a mountain have to be for the pressure atop it to be half the pressure at sea level? Round your answer to 3 significant figures.
$0.317$
$5330$
18. The number of bacteria, $n$, over time, $t$ follows a straight line graph in the form $$\log n = 0.6t + 2$$ Given that $n = ab^t$, find the values of $a$ and $b$.
$a = 100$ and $b = 3.98$
19. Some cities in the UK were ranked by size. Here are their ranks, $r$, and populations, $p$ thousands:
$r$
$2$
$3$
$4$
$5$
$6$
$p$
$1153$
$793$
$582$
$494$
$423$
It is predicted that they are related by the formula $r = ap^b$. By plotting a graph of $\log r$ against $\log p$, estimate the values of $a$ and $b$.
Linear graph has gradient $-0.92$ and intercept $3.34$ $a = 10^{3.34} = 2190$ and $b = $-0.92$
20. Amy is investigating the number of people who fall sick at her school, $n$, over time, $t$ days. She plots a graph of $\log n$ against $t$ which gives a straight line through the points $(0,1.6)$ and $(10,2.55)$ Amy suggests $n = ab^t$ as a model.
Predict the number of sick people after 30 days to 3 significant figures.
Is this an over or under estimate?
$28200$
Overestimate
21. Here are some data which fit the trend $y = ax^b$:
$x$
$3$
$5$
$8$
$10$
$15$
$y$
$16.3$
$33.3$
$64.3$
$87.9$
$155.1$
Plot a suitable straight line graph to estimate the values of $a$ and $b$.
$a = 3.5$ and $b = 1.4$
22. Here are some data which fit the trend $y = ab^x$:
$x$
$2$
$2.5$
$3$
$3.5$
$4$
$y$
$12$
$17$
$24$
$34$
$48$
Plot a suitable straight line graph to estimate the values of $a$ and $b$.
$a = 3$ and $b = 2$
23. The surface area, $s$ and volume, $v$ of some spheres were measured:
$s$
$18$
$50$
$113$
$222$
$314$
$v$
$7$
$33$
$113$
$310$
$523$
Given that $s = av^b$, plot a suitable straight line graph to find the values of $a$ and $b$.
$a = 0.09$ and $b = 1.5$
24. Emily measures the width, $w$, and area, $A$, of some similar shapes. She believes there is a relationship in the form $A = aw^b$ between them, where $a$ and $b$ are constants. When she plots a graph of $\log A$ against $\log w$, she fits a straight line with gradient $2$ and vertical intercept $-0.1049$. What shape is Emily investigating?
