P3 Chapter 5 :: Exponentials & Logarithms
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Chapter Overview
1:: Sketch exponential graphs.
2:: Use an interpret models that use exponential functions.
4:: Understand the log function and use laws of logs.
NEW! to A Level 2017
Again, moved from Year 2.
5:: Use logarithms to estimate values of constants in non-linear models.
(This is a continuation of (2))
-2 -1 1 2 3 4 5 6 7
8
20
16
12
8
4
-4
| -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| 0.25 | 0.5 | 1 | 2 | 4 | 8 | 16 |
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Contrasting exponential graphs
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Graph Transformations
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Exercise 5A
Pearson Pure Mathematics 3
Pages 104-105
Function
Gradient
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Compare each exponential function against its respective gradient function. What do you notice?
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Function
Gradient
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Note: This is not a standalone rule but an application of something called the ‘chain rule’, which you will encounter in Year 2.
Note: In general, when you scale the function, you scale the derivative/integral.
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More Graph Transformations
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Test Your Understanding
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Exercise 5B
Pearson Pure Mathematics Year 1/AS
Pages 107-108
Just for your interest…
Its value was originally encountered by Bernoulli who was solving the following problem:
You have £1. If you put it in a bank account with 100% interest, how much do you have a year later? If the interest is split into 2 instalments of 50% interest, how much will I have? What about 3 instalments of 33.3%? And so on…
No. Instalments | Money after a year |
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*You can find a full proof here in my Graph Sketching/Limits slides: http://www.drfrostmaths.com/resources/resource.php?rid=163
Application 1: Solutions to many ‘differential equations’.
Application 2: Russian Roulette
Application 3: Secret Santa
A scene from one of Dr Frost’s favourite films, The Deer Hunter.
ABC,
ACB,
BAC,
BCA,
CAB,
CBA
Exponential Modelling
What is the initial population?
What is the initial rate of population growth?
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Another Example
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b
c
d
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Exercise 5E
Pearson Pure Mathematics 3
Pages 117-118
-2 -1 1 2 3 4 5 6 7
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4
3
2
1
-1
-2
| 0.25 | 0.5 | 1 | 2 | 4 | 8 |
| -2 | -1 | 0 | 1 | 2 | 3 |
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Root is 1.
The gradient gradually decreases but remains positive (log is an “increasing function”)
Laws of Logs
i.e. You can move the power to the front.
The logs must have a consistent base.
Anti Laws
These are NOT LAWS OF LOGS, but are mistakes students often make:
There is no method to simplify the log of a sum, only the sum of two logs!
🗶 FAIL
🗶 FAIL
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Solving Equations with Logs
This is a very common type of exam question.
The strategy is to combine the logs into one and isolate on one side.
We’ve used the laws of logs to combine them into one.
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Test Your Understanding
Edexcel C2 Jan 2013 Q6
Those who feel confident with their laws could always skip straight to this line.
These are both valid solutions when substituted into the original equation.
a
b
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Solving equations with exponential terms
This is often said “Taking logs of both sides…”
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Solving equations with exponential terms
Logs in general are great for solving equations when the variable is in the power, because laws of logs allow us to move the power down.
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It doesn’t matter what base you use to get the final answer as a decimal, provided that it’s consistent. You may as well use the calculator’s ‘log’ (no base) key.
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Test Your Understanding
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3
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Natural Logarithms
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In previous chapters we’ve already dealt with quadratics in disguise, e.g. “quadratic in sin”. We therefore just apply our usual approach: either make a suitable substitution so the equation is then quadratic, or (strongly recommended!) go straight for the factorisation.
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Test Your Understanding
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Exercise 5C
Pearson Pure Mathematics 3
Page 109-110
Graphs for Exponential Data
In Science and Economics, experimental data often has exponential growth, e.g. bacteria in a sample, rabbit populations, energy produced by earthquakes, my Twitter followers over time, etc.
Because exponential functions increase rapidly, it tends to look a bit rubbish if we tried to draw a suitable graph:
Take for example “Moore’s Law”, which hypothesised that the processing power of computers would double every 2 years. Suppose we tried to plot this for computers we sampled over time:
1970
1980
1990
2000
Number of transistors
Year
If we tried to force all the data onto the graph, we would end up making most of the data close to the horizontal axis. This is not ideal.
1970
1980
1990
2000
Year
log(transistors)
Graphs for Exponential Data
Because the energy involved in earthquakes decreases exponentially from the epicentre of the earthquake, such energy values recorded from different earthquakes would vary wildly.
The Richter Scale is a logarithmic scale, and takes the log (base 10) of the amplitude of the waves, giving a more even spread of values in a more sensible range.
(The largest recorded value on the Richter Scale is 9.5 in Chile in 1960, and 15 would destroy the Earth completely – evil scientists take note)
The result is that an earthquake just 1 greater on the Richter scale would in fact be 10 times as powerful.
Richter Scale
Other Non-Linear Growth
We would also have similar graphing problems if we tried to plot data that followed some polynomial function such as a quadratic or cubic.
We will therefore look at the process to convert a polynomial graph into a linear one, as well as a exponential graph into a linear one…
Turning non-linear graphs into linear ones
Example
Recall that the coefficient of an exponential term gives the ‘initial value’.
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Example
| 0.30 | 0.48 | 0.60 | 0.70 | 0.78 |
| 6 | 5.86 | 5.79 | 5.72 | 5.68 |
6.4
6.0
5.6
5.2
0.2 0.4 0.6 0.8
Let’s use these points on the line of best fit to determine the gradient.
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Test Your Understanding
| 0.7 | 1.3 | 2.2 |
| 3.372 | 3.565 | 3.855 |
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Exercise 5D
Pearson Pure Mathematics 3
Page 113-114
The End