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.
= 8
You know that = 8,
so this equation can be solved by your spotting the answer: x = 3.
= 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:
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.]
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.
Ex 7a
Exponential growth and decay
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 =
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!
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.
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
and therefore produces an arbitrary constant.
All first order equations will produce a single arbitrary constant.…read more

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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.
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
Take logs of bs
ln y = x ln 5
Exponentiate bs
y= = .…read more


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