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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…

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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.…

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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…

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i.e. = km
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,…

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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…

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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 =…

Page 7

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Different formats for exponentials
Take logs of bs
ln y = x ln 5
Exponentiate bs
y= = . =5
Ex 7c
Ex 7d

T3 Compound angle formulae

CT Training 22/02/2010


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