exponentials and logarithms

Explanation of techniques and reference to Q's in C3&C4 Book

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W7.1 Exponential growth and decay and log equations
The first topic is a consequence of our use of natural logarithms.
Solving equations with logs
The main usefulness of logs in solving equations is with exponential equations
­
equations in which the unknown is in the exponent.
Ex
= 8
You know that = 8,
so this equation can be solved by your spotting the answer: x = 3.
Ex
= 7
This is rather more awkward.
You could spot that the answer will be a bit less than 3,
But the only way you know of locating it more precisely would be trial and
improvement [or decimal search or interval bisection or whatever name
you have been told to give it].
But we're now in a position to improve on that.
Take logs of both sides
log( ) = log 7
The third law of logs changes this to:
x log 2 = log 7
This is a simple linear equation in which we divide both sides by log 2:
x=
It's important to note that this has nothing to do with the second law of
logs ­ it's just one nasty number divided by another nasty number.
So it gives x = 2.81 (3 s.f.)
In this example, we've used base 10 logs.
It would work just as easily using natural logs.
[So prove to yourself that gives the same answer.]
N.B.
If the equation you're solving has an exponent of e,
then use natural logs.
e.g. = 4
ln( ) = ln 4
Then, because and ln x are inverse functions of each other,
the LHS is just 3x.
CT Training 22/02/2010

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Similarly, if the exponent is a power of 10, you should use logs base 10.
In all other examples, it doesn't matter which kind of log you use.
Drill
Ex 7a
Exponential growth and decay
Ex
A quantity y decays in proportion to its size for the time being.
Express y as a function of time.
The data: y
[The minus sign denotes that the change is negative ­
y is reducing in size.]
i.e. = ky, for some positive k.…read more

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Another is that the rate of decay of radioactivity of a substance like uranium is
proportional to the current level of radioactivity.
It can also work with growth:
A colony of bacteria, given plenty of room
and plenty of nutrient,
will grow in proportion to its size.
So y =
Drill
p 280 Ex 7b
W7.2 Creating differential equations
A differential equation is an equation containing x, y and [and possibly
higher derivatives too].…read more

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This will give m = A .
Your syllabus allows questions to be asked ­ rather silly questions to my mind ­
in which they give you the solution to a differential equation and ask you what
the question was!
Ex
By eliminating the arbitrary constant, A, find a first order differential equation
that is equivalent to + = Ay.…read more

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A differential equation, by contrast, is not looking for numerical values.
It's looking to find a function y = y(x) which works in the equation.
Ex
=
Technically, this is a differential equation,
but solving it is easy ­ just integrate b.s. w.r.t. x.
[You've been doing that for the last year].
y= + c
[Notice that, in this case, solving the first order equation involves an
integration
and therefore produces an arbitrary constant.
All first order equations will produce a single arbitrary constant.…read more

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Drill
Ex 8b
Differential equations can often come packaged with boundary conditions ­
numerical values of the variables at some stage in the process.
If the stage is the beginning of the process, these boundary conditions are
known as initial conditions.
WEx
x = y + 3; y = 3 when x = 2.
This gives ln |y + 3| = ln x + c.…read more

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Different formats for exponentials
y=
Take logs of bs
ln y = x ln 5
Exponentiate bs
y= = .…read more

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