P1 Chapter 14 :: Exponentials & Logarithms
Transcript of P1 Chapter 14 :: Exponentials & Logarithms
P1 Chapter 14 :: Exponentials & Logarithms
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Last modified: 28th August 2017
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Chapter Overview
1:: Sketch exponential graphs. 2:: Use an interpret models that use exponential functions.
You have encountered exponential expressions like π¦ = 2π₯ before, but probably not βtheβ exponential function π¦ = ππ₯. Similarly, you will learn that the inverse of π¦ = 2π₯ is π¦ = log2 π₯.
The population π of Davetown after π‘ years is modelled using π = π΄πππ‘, where π΄, π are constantsβ¦
Sketch π¦ = 2π₯ and π¦ = ππ₯ on the same axes.
4:: Understand the log function and use laws of logs.
Solve the equation:log2 2π₯ = log2 5π₯ + 4 β 3
NEW! to A Level 2017βTheβ exponential function, ππ₯, has been moved from Year 2 to Year 1.
NEW! to A Level 2017Again, moved from Year 2.
5:: Use logarithms to estimate values of constants in non-linear models.
(This is a continuation of (2))
3:: Be able to differentiate πππ₯.
If π¦ = 5π3π₯, determine ππ¦
ππ₯.
-2 -1 1 2 3 4 5 6 7
8
20
16
12
8
4
-4
π -2 -1 0 1 2 3 4
π 0.25 0.5 1 2 4 8 16π¦ = 2π₯
Why are exponential functions important?Each of the common graphs have a key property that makes them useful for modelling.
For reciprocal graphs π¦ =π
π₯, as π₯ doubles, π¦
halves. This means weβd use it to represent variables which are inversely proportional.
Linear graphs are used when weβre adding the same amount each time.In contrast, exponential graphs are used when weβre multiplying by the same amount each time. For example, we might use 1000(1.05π‘) to model our savings with interest, where each year we have 1.05 times as much, i.e. with 5% added interest.
Fro Note: Ensure that you can distinguish between a π₯π (e.g. polynomial) term and an ππ₯ exponential term. In the former the variable is in the base, and in the letter the variable is in the power. 2π₯ behaves very differently to π₯2, both in its rate of growth (i.e. exponential terms grow much faster!) and how it differentiates.
Contrasting exponential graphs
On the same axes sketch π¦ = 3π₯, π¦ = 2π₯, π¦ = 1.5π₯
On the same axes sketch π¦ = 2π₯ and π¦ =1
2
π₯
π₯
π¦π = ππ
π = ππ
π = π. ππ
The π¦-intercept is always 1.
If π₯ < 0, the larger the base, the smaller the π¦ value:3π₯ < 2π₯ < 1.5π₯
If π₯ > 0, the larger the base, the larger the π¦ value:
1.5π₯ < 2π₯ < 3π₯
1
π₯
π¦
π = πππ =π
π
π
Three important notes:β’ π¦ = 2π₯ is said to be βexponential growingβ whereas
π¦ =1
2
π₯is said to be βexponentially decayingβ,
because itβs getting smaller (halving) each time π₯increases by 1.
β’ π¦ =1
2
π₯is a reflection of π¦ = 2π₯ in the line π₯ = 0.
Proof:
If π π₯ = 2π₯, π βπ₯ = 2βπ₯ =1
2π₯=
1
2
π₯
β’1
2
π₯would usually be written 2βπ₯.
You should therefore in general be able to recognise and sketch the graph π = πβπ.
1
Graph Transformations
Sketch π¦ = 2π₯+3
π₯
π¦
π = ππ+π
8
The βchangeβ to 2π₯ is βinside the functionβ (i.e. input π₯ is replaced with π₯ + 3).So a translation to the left by 3.
Ensure you work out the new π¦-intercept.
Exercise 14A
Pearson Pure Mathematics Year 1/ASPages 313-314
You wonβt yet be able to differentiate general exponential functions π¦ = ππ₯ until Year 2.
But Iβve calculated the gradient functions for you β click the black arrow to reveal the graph and gradient
function.
π¦ = ππ₯
Function Gradient
π¦ = 1π₯
π¦ = 1.5π₯
π¦ = 2π₯
π¦ = 2.5π₯
π¦ = 3π₯
π¦ = 3.5π₯
ππ¦
ππ₯= 0
ππ¦
ππ₯= 0.41 Γ 1.5π₯
>
>
ππ¦
ππ₯= 0.69 Γ 2π₯ >
ππ¦
ππ₯= 0.92 Γ 2.5π₯ >
ππ¦
ππ₯= 1.10 Γ 3π₯ >
ππ¦
ππ₯= 1.25 Γ 3.5π₯ >
Compare each exponential function against its respective gradient function. What do you notice?
π¦ = ππ₯
Function Gradient
π¦ = 1π₯
π¦ = 1.5π₯
π¦ = 2π₯
π¦ = 2.5π₯
π¦ = ππ₯ππ¦
ππ₯= ππ₯
ππ¦
ππ₯= 0
ππ¦
ππ₯= 0.41 Γ 1.5π₯
>
>
ππ¦
ππ₯= 0.69 Γ 2π₯ >
ππ¦
ππ₯= 0.92 Γ 2.5π₯ >
>π¦ = 3π₯ππ¦
ππ₯= 1.10 Γ 3π₯ >
π¦ = 3.5π₯ππ¦
ππ₯= 1.25 Γ 3.5π₯ >
π¦ = 2.5π₯ and π¦ = 3π₯ seem to be similar to their respective gradient functions. So is there a base between 2.5 and 3 where the function is equal to its gradient function?
π = 2.71828β¦ is known as Eulerβs Number.
It is one of the five most fundamental constants in mathematics (0, 1, π, π, π).
It has the property that:
π¦ = ππ₯ βππ¦
ππ₯= ππ₯
Although any function of the form π¦ = ππ₯ is known as an exponential function, ππ₯ is known as βtheβ exponential function.
You can find the exponential function on your calculator, to the right (above the βlnβ key)
Differentiating π¦ = ππππ₯
! If π¦ = πππ₯, where π is a constant, then ππ¦
ππ₯= ππππ₯
Different π5π₯ with respect to π₯.
ππ¦
ππ₯= 5π5π₯
Note: This is not a standalone rule but an application of something called the βchain ruleβ, which you will encounter in Year 2.
Different πβπ₯ with respect to π₯.
π¦ = πβ1π₯ β΄ππ¦
ππ₯= βπβπ₯
Different 4π3π₯ with respect to π₯.
ππ¦
ππ₯= 12π3π₯
Note: In general, when you scale the function, you scale the derivative/integral.
More Graph Transformations
Sketch π¦ = π3π₯
π₯
π¦
1
Input π₯ is being replaced with 3π₯, so stretch of scale
factor 1
3on π₯-axis.
Sketch π¦ = 5πβπ₯
π = πππ
π = ππ
π₯
π¦
1
Note: Recall the shape of a π¦ = πβπ₯ graph.
5 is βoutsideβ function, so stretch of scale factor 5 on π¦-axis.
π = ππβπ
π = ππ
5
Sketch π¦ = 2 + π1
3π₯
π₯
π¦
1
We have a stretch on π₯-axis by scale factor 3, and a translation up by 2.
π = π + ππππ
π = ππ
3
Important Note: Because the original asymptote was π¦ = 0, it is now π¦ = 2 and you must indicate this along with its equation.
π = π
Test Your Understanding
Sketch π¦ = πβ2π₯ β 1
π₯
π¦
π = πβππ β π
π¦ = πβ2 0 β 1 = 0
π = βπ
Exercise 14B
Pearson Pure Mathematics Year 1/ASPages 316-317
Just for your interestβ¦
Where does π come from, and why is it so important?
π = 2.71828β¦is known as Eulerβs Number, and is considered one of the five fundamental constants in maths: 0, 1, π, π, π
Its value was originally encountered by Bernoulli who was solving the following problem:You have Β£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 onβ¦
No. Instalments Money after a year
1 1 Γ 21 = Β£2
2 1 Γ 1.52 = Β£2.25
3 1 Γ 1. αΆ33 = Β£2.37
4 1 Γ 1.254 = Β£2.44
π1 +
1
π
π
As π becomes larger, the amount after a year approaches Β£2.71β¦, i.e. π!
Thus:
π = limπββ
1 +1
π
π
But we have seen that differentiation by first principles uses βlimitsβ. It is therefore possible to prove from the definition
above that π
ππ₯ππ₯ = ππ₯ , and these two definitions of π
are considered to be equivalent*.
π therefore tends to arise in problems involving limits, and also therefore crops up all the time in anything involving differentiation and integration. Letβs see some applicationsβ¦
*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 βdifferential equationsβ.
Frequently in physics/maths, the rate of change of a variable is proportional to the value itself. So with a population πbehaving in this way, if the population doubled, the rate of increase would double.
π βππ
ππ‘β π = π
ππ
ππ‘
This is known as a βdifferential equationβ because the equation involves both the
variable and its derivative ππ
ππ‘.
The βsolutionβ to a differentiation equation means to have an equation relating π and
π‘ without the ππ
ππ‘. We end up with (using
Year 2 techniques):
π· = π¨πππ
where π΄ and π are constants. This is expected, because we know from experience that population growth is usually exponential.
π
π‘
Application 2: Russian Roulette
Application 3: Secret Santa
I once wondered (as you do), if I was playing Russian Roulette, where you randomly rotate the barrel of a gun each time with π chambers, but with one bullet, whatβs the probability Iβm still alive after π shots?
The probability of surviving each time is
1 β1
π, so the probability of surviving all π shots is 1 β
1
π
π. We might consider what
happens when π becomes large, i.e. limπββ
1 β1
π
π. In general, ππ = lim
πββ1 +
π
π
π.
Thus limπββ
1 β1
π
π= πβ1 =
1
π, i.e. I have a 1 in π chance of surviving. Bad odds!
This is also applicable to the lottery. If there was a 1 in 20 million chance of winning the lottery, we might naturally wonder what happens if we bought 20 million (random) lottery tickets. Thereβs a 1 in π (roughly a third) chance of winning no money at all!
A scene from one of Dr Frostβs favourite films, The Deer Hunter.
You might have encountered π! = π Γ π β 1 Γβ―Γ 2 Γ 1, said βπ factorialβ meaning βthe number of ways of arranging πobjects in a lineβ. So if we had 3 letters ABC, we have 3! = 6ways of arranging them.
Meanwhile, ! π means the number of derangements of π, i.e. the arrangements where no letter appears in its original place.
ABC, ACB, BAC, BCA, CAB, CBA
For ABC, that only gives BCA or CAB, so ! 3 = 2. This is applicable to βSecret Santaβ (where each person is given a name out a hat of whom to give their present to) because ideally we want the scenario where no person gets their own name.
Remarkably, a derangement occurs an π-th of the time. So if there are 5 people and hence 5! =120 possible allocations of recipient names, we only get the ideal Secret Santa situation just 120
π= 44.15 β 44 times. And so we get my
favourite result in the whole of mathematics:
(where [β¦ ]means round)! π =
π!
π
Exponential Modelling
There are two key features of exponential functions which make them suitable for population growth:
1. ππ gets π times bigger each time π increases by 1. (Because ππ+π = π Γ ππ)With population growth, we typically have a fixed percentage increase each year. So suppose the growth was 10% a year, and we used the equivalent decimal multiplier, 1.1, as π. Then 1.1π‘, where π‘ is the number of years, would get 1.1 times bigger each year.
2. The rate of increase is proportional to the size of the population at a given moment.This makes sense: The 10% increase of a population will be twice as large if the population itself is twice as large.
π
π‘
Suppose the population π in The Republic of Dave is modelled by π = 100π3π‘
where π‘ is the numbers years since The Republic was established.
What is the initial population? When π = π,π· = πππππ = πππ
What is the initial rate of population growth?
π π·
π π= ππππππ
When π = π,π π·
π π= πππ
βUse of Technologyβ Monkey says:When Iβm not busy eating ticks off other monkeys, I use the πβ‘ key. I can also use [ALPHA] [π] to get π without a power.
[Textbook] The density of a pesticide in a given section of field, π mg/m2, can be modelled by the equation π = 160πβ0.006π‘
where π‘ is the time in days since the pesticide was first applied.a. Use this model to estimate the density of pesticide after 15 days.b. Interpret the meaning of the value 160 in this model.
c. Show that ππ
ππ‘= ππ, where π is a constant, and state the value of π.
d. Interpret the significance of the sign of your answer in part (c).e. Sketch the graph of π against π‘.
Another Example
a When π‘ = 15, π = 160πβ0.006(15)
= 145.2 ππ/π2
When π‘ = 0, then π = 160. Thus 160 is the initial density of pesticide in the field.
ππ
ππ‘= 160 Γ β0.006 πβ0.006π‘
= β0.96πβ0.006π‘ β΄ π = β0.96
The rate is negative, thus the density of pesticide is decreasing.
In general, the βinitial valueβ, when π‘ = 0, is the coefficient of the exponential term.
b
c
d
π
π‘
160
Recall that if the power in the exponential term is negative, we have an exponential decay graph, which gradually decreases towards 0.If in doubt, just try a large value of π‘.Donβt forget to put in the 160!
Exercise 14C
Pearson Pure Mathematics Year 1/ASPages 318-319
Logarithms
4 π₯ Γ 3 π₯ Γ· 3 4
Function Inverse
4 π₯ + 3 π₯ β 3 4
4 π₯2 π₯ 4
4 π₯5 5 π₯ 4
12
7
16
1024
4 3π₯ log3 π₯ 481
You know the inverse of many mathematical operations; we can undo an addition by 2 for example by subtracting 2. But is there an inverse function for an exponential function?
Such functions are known as logarithms, and exist in order to provide an inverse to exponential functions.
Interchanging between exponential and log form
32 = 9 log3 9 = 2
! logπ π (βsaid log base π of πβ) is equivalent to ππ₯ = π.The log function outputs the missing power.
Here are two methods of interchanging between these forms.Pick your favourite!
Method 1: βMissing Powerβ
β’ Note first the base of the log must match the base of the exponential.
β’ log2 8 for example asks the question β2 to what power gives 8?β
Click to start Fro-manimation
πππ 82
3== 3
We can imagine inserting the output of the log just after the base. Click the button!
Method 2: Do same operation to each side of equation.
Since KS3 youβre used to the idea of doing the same thing to each side of the equation that βundoesβ whatever you want to get rid of.
3π₯ + 2 = 11
3π₯ = 9β2 β2
βlog base πβ undoes βπ to the power ofβ and vice versa, as they are inverse functions.
log2 8 = 3
8 = 232β‘ 2β‘
Examples
log5 25 = πThink: β5 to the power of what gives you 25?β
log3 81 = π
log4 1 = π
log2 32 = π
log4 4 = π
log21
2= βπ
log31
27= βπ
log21
16= βπ
log4 β1 =
log10 1000 = π logπ π3 = π! logπ 1 = 0for all π.
While a log can output a negative number, we canβt log negative numbers.
Strictly Just For Your Interest: However, if we were to expand the range (i.e. output) of the log function to allow complex numbers (known as the βcomplex logarithmβ), then we in fact get
log4(β1) =ππ
ln 4. Itβs probably better if you purge these last few
sentences from your memory and move alongβ¦
NOPE?!!??
With Your Calculator
There are three buttons on your calculator for computing logs:
logβ‘ β‘log3 7 = π. πππππβ¦log5 0.3 = βπ. πππππβ¦
ππln 10 = π. πππππβ¦ln π = π
I couldnβt think of a word that rhymed with βlnβ so I recorded it for you.
ππ is the βnatural log of πβ, meaning βlog to the base πβ, i.e. it the inverse of ππ₯.
ln π₯ = logπ π₯We will use it more extensively later this chapter.
Just like the β symbol without a number is 2 β‘ by default, πππ without a base is base 10 by default when used on your calculator (although confusingly βπππβ can mean βππβ in mathematical papers)
πππ log 100 = π
ln of π₯
Exercise 14D
Pearson Pure Mathematics Year 1/ASPages 320-321
[MAT 2015 1J] Which is the largest of the following numbers?
A) 7
2B)
5
4C)
10!
3 6!
D) log2 30
log3 85E)
1+ 6
3
(Official solution) Squaring all answers
results in (A) π
πwhich is larger than (B)
ππ
ππ.
After squaring (C) it simplifies to ππΓπΓπΓπ
πΓπ!
which further simplifies to π
πwhich is smaller
than (A). π₯π¨π π ππ β π and π₯π¨π π(ππ) β π, hence (D) is smaller than (A) after squaring. Comparing (A) with (E) after squaring results in a
comparison of π
πand
π+π π
π. As π < π < π,
(E) squared must be less than ππ
πand hence
less than π
π. The answer is (A).
1
Extension
Non-calculator!
[MAT 2013 1F] Three positive numbers π, π, πsatisfy
logπ π = 2logπ π β 3 = 3logπ π + 5 = 2
This information:A) specifies π uniquely;B) is satisfied by two values of π;C) is satisfied by infinitely many values of π;D) is contradictory
If we take exponents of the three equations:π = ππ, π β π = ππ, π + π = ππ
Hence eliminating π and π we getππ + π = ππ β π β ππ π β π = π
We are only interested in positive solutions to π. Note that ππ π β π is negative for π < π < π and then both ππ and π β π are positive and increasing for π > π. So there is only one positive solution to the equation (π = π, so that π = π, π = ππ). Answer is (a).
2
-2 -1 1 2 3 4 5 6 7
8
4
3
2
1
-1
-2
π 0.25 0.5 1 2 4 8
π -2 -1 0 1 2 3π¦ = log2 π₯
The log graph isnβt defined for π₯ β€ 0
We have a vertical asymptote π₯ = 0
Root is 1.
The gradient gradually decreases but remains positive (log is an βincreasing functionβ)
Laws of Logs
! Three main laws:logπ π₯ + logπ π¦ = logπ π₯π¦
logπ π₯ β logπ π¦ = logππ₯
π¦
logπ π₯π = π logπ π₯
Special cases:logπ π = 1 (π > 0, π β 1)logπ 1 = 0 π > 0, π β 1
log1
π₯= log π₯β1 = β log π₯
Not in syllabus (but in MAT/PAT):
logπ π =logπ π
logπ π
i.e. You can move the power to the front.
We often try to avoid leaving fractions inside logs.So if the answer was:
log21
3
You should write your answer as: β log2 3Reciprocating the input negates the output.
The logs must have a consistent base.
This is known as changing the base. So to get log2 9 in terms of log base 3:
πππ29 =πππ39
πππ32=
2
πππ32
Examples
Write as a single logarithm:a. log3 6 + log3 7b. log2 15 β log2 3c. 2 log5 3 + 3 log5 2
d. log10 3 β 4 log101
2
Write in terms of logπ π₯, logπ π¦ and logπ π§a. logπ(π₯
2π¦π§3)
b. logππ₯
π¦3
c. logππ₯ π¦
π§
d. logππ₯
π4
log3 42
log2 5
log5 9 + log2 8 = log5 72
log10 3 β log101
16= log1048
a
b
c
d
logπ(π₯2π¦π§3) = logπ(π₯
2) + logπ π¦ + logπ π§3
= 2 logπ π₯ + logπ π¦ + 3 logπ π§
logππ₯
π¦3= logπ(π₯) β logπ π¦3
= logπ π₯ β 3 logπ π¦
logπ π₯π¦12 β logπ π§ = logπ π₯ + logπ π¦
12 β logπ π§
= logπ π₯ +1
2logπ
1
2β logπ π§
logππ₯
π4= logπ π₯ β 4 logπ π = logπ π₯ β 4
a
b
c
d
Anti Laws
These are NOT LAWS OF LOGS, but are mistakes students often make:
logπ π + π = logπ π + logπ πThere is no method to simplify the log of a sum, only the sum of two logs!
log2 π₯3 = 3 log2 π₯ The power must be on the input
(here the π₯), but here the power is around the entire log.
FAIL
FAIL
Solving Equations with Logs
Solve the equation log10 4 + 2 log10 π₯ = 2
This is a very common type of exam question.The strategy is to combine the logs into one and isolate on one side.
log10 4 + log10 π₯2 = 2
log10 4π₯2 = 24π₯2 = 102
π₯2 = 25π₯ = 5
Weβve used the laws of logs to combine them into one.
Use your favourite method of rearranging. Either do β10 the power of each sideβ to βundoβ the log, or the βinsert the 2 between the 10 and 4π₯2β method.
The subtle bit: You must check each value in the original equation. If π₯ = β5, then weβd have log10 β5 but weβre not allowed to log a negative number.
Test Your Understanding
Edexcel C2 Jan 2013 Q6
log2 π₯ + 15 2 β log2 π₯ = 6
log2π₯ + 15 2
π₯= 6
26 =π₯ + 15 2
π₯64π₯ = π₯ + 15 2
64π₯ = π₯2 + 30π₯ + 225π₯2 β 34π₯ + 225 = 0
π₯ β 25 π₯ β 9 = 0π₯ = 25 ππ π₯ = 9
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
Exercise 14E
Pearson Pure Mathematics Year 1/ASPages 323-324
Extension
[AEA 2010 Q1b] Solve the equation
log3 π₯ β 7 β1
2log3 π₯ = 1 β log3 2
[AEA 2008 Q5i] Anna, who is confused about the rules of logarithms, states that
log3 π2 = log3(π
2)log3 π + π = log3 π + log3 π
However, there is a value for π and a value for π for which both statements are correct. Find their values.
[MAT 2007 1I] Given that π and π are positive and
4 log10 π2 + log10 π
2 = 1what is the greatest possible value of π?
[MAT 2002 1F] Observe that 23 = 8, 25 = 32, 32 = 9 and 33 = 27. From these facts, we can deduce that log2 3 is:
A) between 11
3and 1
1
2
B) between 11
2and 1
2
3
C) between 12
3and 2
D) none of the above
These are all strictly non-calculator!
Solutions on next slide.
1
2
3
4
Solutions to Extension Exercises
[AEA 2010 Q1b] Solve the equation
log3 π₯ β 7 β1
2log3 π₯ = 1 β log3 2
Solution: π =ππ
π
[AEA 2008 Q5i] Anna, who is confused about the rules of logarithms, states that
log3 π2 = log3(π
2)log3 π + π = log3 π + log3 π
However, there is a value for π and a value for π for which both statements are correct. Find their values.First equation:
π₯π¨π π ππ = π π₯π¨π π π
β΄ π₯π¨π π π π₯π¨π π π β π = ππ₯π¨π π π β π = ππ₯π¨π π π = π β π = π
Second equation:π₯π¨π π π + π = π₯π¨π π ππ
β΄ π + π = ππ β π =π
π β πβ΄ π β π
π = π, π =π
π
[MAT 2007 1I] Given that π and π are positive and4 log10 π
2 + log10 π2 = 1
what is the greatest possible value of π?
To make π as large as possible we make π₯π¨π ππ π
π as small as possible. Anything squared is at least 0: π = π will achieve this.
β΄ π π₯π¨π ππ ππ = π
π₯π¨π ππ π = Β±π
ππ = ππ
[MAT 2002 1F] Observe that 23 = 8, 25 = 32, 32 = 9 and 33 = 27. From these facts, we can deduce that log2 3 is:
A) between 11
3and 1
1
2
B) between 11
2and 1
2
3
C) between 12
3and 2
D) none of the above
Suppose that π₯π¨π π π >π
π. Taking 2 to the power of each
side:
π > πππ β ππ > ππ β π > π
This is true, so answer is not (A).
Next try π₯π¨π π π >π
π
π > πππ β ππ > ππ β ππ > ππ
This is not true, so answer is (B).
1
2
3
4
Solving equations with exponential terms
Solve 3π₯ = 20
Applying πππ3 to each side of the equation:π₯ = log3 20 = 2.727 (π‘π 3ππ)
Solve 54π₯β1 = 61
Applying πππ5 to each side of the equation:4π₯ β 1 = log5 61
π₯ =log5 61 + 1
4= 0.889 (π‘π 3ππ)
This is often said βTaking logs of both sidesβ¦β
Solving equations with exponential terms
Solve 3π₯ = 2π₯+1
Why can we not apply quite the same strategy here?Because the exponential terms donβt have the same base, so we canβt apply the same log.
We βtake logs ofβ/apply log to both sides, but we need not specify a base. πππ on its own may either mean πππ10 (as per your calculator) or ππππ (in academic circles, as well as on sites like WolframAlpha), but the point is, the base does not matter, provided that the base is consistent on both sides.
log 3π₯ = log2π₯+1
π₯ log 3 = π₯ + 1 log 2π₯ log 3 = π₯ log 2 + log 2π₯ log 3 β π₯ log 2 = log 2π₯ log 3 β log 2 = log 2
π₯ =log 2
log 3 β log 2= 1.7095
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.
This then becomes a GCSE-style βchanging the subjectβ type question. Just isolate π₯ on one side and factorise out.
It doesnβt matter what base you use to get the final answer as a decimal, provided that itβs consistent. You may as well use the calculatorβs βlogβ (no base) key.
Test Your Understanding
Solve 32π₯β1 = 5, giving your answer to 3dp.
Solve 2π₯3π₯+1 = 5, giving your answer in exact form.
2π₯ β 1 = log3 5
π₯ =log3 5 + 1
2= 1.232
log 2π₯3π₯+1 = log5log 2π₯ + log3π₯+1 = log 5π₯ log 2 + π₯ + 1 log 3 = log 5π₯ log 2 + π₯ log 3 + log 3 = log 5π₯ log 2 + log 3 = log 5 β log 3
π₯ =log 5 β log 3
log 2 + log 3which could be simplified to:
π₯ =log 5 β log 3
log 6
Solve 3π₯+1 = 4π₯β1, giving your answer to 3dp.
log 3π₯+1 = log 4π₯β1
π₯ + 1 log 3 = π₯ β 1 log 4π₯ log 3 + log 3 = π₯ log 4 β log 4π₯ log 4 β π₯ log 3 = log 3 + log 4
π₯ log 4 β log 3 = log 3 + log 4
π₯ =log 3 + log 4
log 4 β log 3= 8.638
1 3
2
Exercise 14F
Pearson Pure Mathematics Year 1/ASPage 325
Extension
[MAT 2011 1H] How many positive values π₯which satisfy the equation:
π₯ = 8log2 π₯ β 9log3 π₯ β 4log2 π₯ + log0.5 0.25
π = πππ₯π¨π π π
β πππ₯π¨π π π
β πππ₯π¨π π π
+ π
π = ππ π₯π¨π π π β ππ π₯π¨π π π β ππ π₯π¨π π π + π
π = ππ₯π¨π π ππβ ππ₯π¨π π π
πβ ππ₯π¨π π π
π+ π
π = ππ β ππ β ππ + πππ β πππ β π + π = πππ π β π β π π β π = π
ππ β π π β π = π
π + π π β π π β π = πThis has 2 positive solutions.
1
2 [MAT 2013 1J] For a real number π₯ we denote by [π₯] the largest integer less than or equal to π₯. Let π be a natural number. The integral
ΰΆ±0
π
2π₯ ππ₯
equals:
(A) log2 2π β 1 !
(B) π 2π β log2 2π !
(C) π 2π
(D) log2 2π !
(Warning: This one really is very challenging, even for MAT)
Solution to Extension Question 2
[MAT 2013 1J] For a real number π₯we denote by [π₯] the largest integer less than or equal to π₯. Let π be a natural number. The integral
ΰΆ±0
π
2π₯ ππ₯
equals:
(A) log2 2π β 1 !
(B) π 2π β log2 2π !
(C) π 2π
(D) log2 2π ! π₯
π¦π¦ = 2π₯
123
4
5
2π
log2 1 log2 2log2 3
log2 4log2 5 log2 2
π
This biggest challenge is sketching the graph! Because of the rounding down, the graph jumps up 1 at a time, giving a bunch of rectangles. We can use logs to find the corresponding π values at which these jumps occur, which progressively become closer and closer together. The last π value is ππ, thus the last π value is π₯π¨π π π
π = π.
The area, using the rectangles, is thus:π π₯π¨π π π β π₯π¨π π π + π π₯π¨π π π β π₯π¨π π π + π π₯π¨π π π β π₯π¨π π π +β―+ ππ β π π₯π¨π π π
π β π₯π¨π π ππ β π= π₯π¨π π πβ π₯π¨π π π + π π₯π¨π π π β π π₯π¨π π π + π π₯π¨π π π β π π₯π¨π π π +β―+ ππ β π π₯π¨π π π
π β ππ β π ππππ(ππβπ)
= β π₯π¨π π π β π₯π¨π π π ββ―β π₯π¨π π ππ β π + ππ β π ππππ ππ
= β(π₯π¨π π π Γ π Γ π Γβ―Γ ππ β π + π ππ β π
= π ππ β ππππ ππ !
Natural Logarithms
We have previously seen that π¦ = logπ π₯ is the inverse function of π¦ = ππ₯.We also saw that ππ₯ is βtheβ exponential function.The inverse of ππ₯ is logπ π₯, but because of its special importance, it has its own function name!
! The inverse of π¦ = ππ₯ is π¦ = ln π₯
ln ππ₯ = ππln π₯ = π
Since βπ to the power ofβ and βππ ofβ are inverse functions, they cancel each other out!
Solve ππ₯ = 5 Solve 2 ln π₯ + 1 = 5
π₯ = ln 5βln both sidesβ. On the LHS it cancels out the βπto the power ofβ
ln π₯ = 2π₯ = π2
Do βπ to the power ofβ each side. On the LHS it cancels out the ln.
Quadratics in ππ₯
In previous chapters weβve already dealt with quadratics in disguise, e.g. βquadratic in sinβ. 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.
Solve π2π₯ + 2ππ₯ β 15 = 0 Solve ππ₯ β 2πβπ₯ = 1
Note that π2π₯ = ππ₯ 2 thereforeππ₯ + 5 ππ₯ β 3 = 0ππ₯ = β5 ππ ππ₯ = 3
Exponential functions are always positive therefore:
π₯ = ln 3
First write negative powers as fractions:
ππ₯ β2
ππ₯= 1
ππ₯ 2 β 2 = ππ₯
ππ₯ 2 β ππ₯ β 2 = 0ππ₯ + 1 ππ₯ β 2 = 0ππ₯ = β1 ππ ππ₯ = 2π₯ = ln 2
Test Your Understanding
Solve ln 3π₯ + 1 = 2
3π₯ + 1 = π2
π₯ =π2 β 1
3
Solve π2π₯ + 5ππ₯ = 6
π2π₯ + 5ππ₯ β 6 = 0ππ₯ + 6 ππ₯ β 1 = 0ππ₯ = β6 ππ ππ₯ = 1π₯ = ln 1 = 0
Solve 2π₯ππ₯+1 = 3 giving your answer as an exact value.
ln 2π₯ππ₯+1 = ln3ln 2π₯ + ln ππ₯+1 = ln3π₯ ln 2 + π₯ + 1 = ln 3π₯ ln 2 + 1 = ln 3 β 1
π₯ =ln 3 β 1
ln 2 + 1
Exercise 14G
Pearson Pure Mathematics Year 1/ASPage 327-328
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 βMooreβs Lawβ, 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
Nu
mb
er o
f tr
ansi
sto
rs
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.
But suppose we took the log of the number of transistors for each computer. Suppose the number of transistors one year was π¦, then doubled 2 years later to get 2π¦.When we log (base 2) these:
π¦ β log2 π¦2π¦ β log2(2π¦) = log2 2 + log2 π¦
= 1 + log2 π¦The logged value only increased by 1! Thus taking the log of the values turns exponential growth into linear growth(because each time we would have doubled, weβre now just adding 1), and the resulting graph is a straight line.
1970 1980 1990 2000Year
log(
tran
sist
ors
)
Graphs for Exponential Data
Because the energy involved in earthquakesdecreases 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 β 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
π₯
π¦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 oneβ¦
π¦ = 2π₯3
Turning non-linear graphs into linear ones
Case 1: Polynomial β Linear Case 2: Exponential β Linear
Suppose our original model was a polynomial one*:
π¦ = ππ₯π
Then taking logs of both sides:log π¦ = log ππ₯π
log π¦ = log π + π log π₯We can compare this against a straight line:
π = ππ + π
* We could also allow non-integer π; the term would then not strictly be polynomial, but weβd still say the function had βpolynomial growthβ.
! If π¦ = ππ₯π, then the graph of log π¦against log π₯ will be a straight line wih gradient π and vertical intercept log π.
log π₯
log π¦
log π
Suppose our original model was an exponential one:π¦ = πππ₯
Then taking logs of both sides:log π¦ = log πππ₯
log π¦ = log π + π₯ log πAgain we can compare this against a straight line:
π = ππ + π
! If π¦ = πππ₯, then the graph of log π¦against π₯ will be a straight line with gradient log π and vertical intercept log π.
π₯
log π¦
log π
The key difference compared to Case 1 is that weβre only logging the πvalues (e.g. number of transistors), not the π₯ values (e.g. years elapsed). Note that you do not need to memorise the contents of these boxes and we will work out from scratch each timeβ¦
Example
[Textbook] The graph represents the growth of a population of bacteria, π, over π‘ hours. The graph has a gradient of 0.6 and meets the vertical axis at 0,2 as shown.A scientist suggest that this growth can be modelled by the equation π = πππ‘, where π and π are constants to be found.a. Write down an equation for the line.b. Using your answer to part (a) or otherwise, find the values of
π and π, giving them to 3 sf where necessary.c. Interpret the meaning of the constant π in this model.
π‘
log(π)
2
logπ = 0.6π‘ + 2 Equation of straight line is π¦ = ππ₯ + π where here:π¦ = logπ ,π = 0.6, π = 2, π₯ = π‘
Just like on previous slide, start with the model then log it:π = πππ‘
log π = log π + π‘ log πComparing with our straight line equation in (a):log π = 2 β π = 102 = 100log π = 0.6 β π = 100.6 = 3.98 3π π
π gives the initial size of the bacteria population.
Recall that the coefficient of an exponential term gives the βinitial valueβ.
a
b
c
Recall that log πmeans log10 π
Example
[Textbook] The table below gives the rank (by size) and population of the UKβs largest cities and districts (London is number 1 but has been excluded as an outlier).
City Bβham Leeds Glasgow Sheffield BradfordRank, πΉ 2 3 4 5 6Population, π· 1 000 000 730 000 620 000 530 000 480 000
The relationship between the rank and population can be modelled by the formula:π = πππ where π and π are constants.a) Draw a table giving values of logπ and logπ to 2dp.b) Plot a graph of logπ against logπ using the values from your table and draw the line of best fit.c) Use your graph to estimate the values of π and π to two significant figures.
πππ πΉ 0.30 0.48 0.60 0.70 0.78
πππ π· 6 5.86 5.79 5.72 5.68
log π
log(π)
6.4
6.0
5.6
5.2
0.2 0.4 0.6 0.8
0.05, 6.16
0.77, 5.68
Letβs use these points on the line of best fit to determine the gradient.
First get equation of straight line:
π =Ξπ¦
Ξπ₯=5.68 β 6.16
0.77 β 0.05= β0.67
π = 6.2 (reading from the graph)β΄ log π = β0.67 log π + 6.2
As with previous example, letβs log the original model so we can compare against our straight line:
π = πππ
logπ = logπ + π logπComparing this with our straight line:
log π = 6.2 β π = 106.2 = 1600000 2π ππ = β0.67
a
b
c
Test Your UnderstandingReflections: Consider what weβre doing in this whole process in case you donβt understand why weβre doing all of this:1. We want to find the parameters
of a model, e.g. π = πππ‘ that best fits the data (in this case the parameters we want to find are πand π).
2. If the data had a linear trend, then this would be easy! We know from KS3 that weβd just plot the data, find the line of best fit, then use the gradient and π¦-intercept to work out the π and π in our linear model.
3. But the original data wasnβt linear, and it would be harder to draw an βexponential curve of best fitβ.
4. We therefore log the model so that the plotted data then roughly forms a straight line, and then we can then draw a (straight) line of best fit.
5. The gradient and π¦-intercept of this line then allows us to estimate the parameters π and πin the original model that best fit the data.
The process of finding parameters in a model, that best fits the data, is known as regression.
Dr Frostβs wants to predict his number of Twitter followers π (@DrFrostMaths) π‘years from the start 2015. He predicts that his followers will increase exponentially according to the model π = πππ‘, where π, π are constants that he wishes to find.
He records his followers at certain times. Here is the data:
Years π after 2015: 0.7 1.3 2.2Followers π·: 2353 3673 7162
a) Draw a table giving values of π‘ and logπ (to 3dp).b) A line of best fit is drawn for the data in your new table, and it happens to go
through the first data point above (where π‘ = 0.7) and last (where π‘ = 2.2).Determine the equation of this line of best fit. (The π¦-intercept is 3.147)
c) Hence, determine the values of π and π in the model.d) Estimate how many followers Dr Frost will have at the start of 2020 (when
π‘ = 5).
π 0.7 1.3 2.2
πππ π· 3.372 3.565 3.855
π =3.855 β 3.372
2.2 β 0.7= 0.322
π = 3.147β΄ logπ = 0.322π‘ + 3.147
π = πππ‘
logπ = logπ + π‘ log πβ΄ logπ = 3.147 β π = 1403log π = 0.322 β π = 2.099
π = 1403 2.099 π‘
When π‘ = 5, π = 57164
a b
c d
Exercise 14H
Pearson Pure Mathematics Year 1/ASPage 331-333
The End