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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…

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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.…

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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!
Ex
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.

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…

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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 =…

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

Test
T3 Compound angle formulae




CT Training 22/02/2010

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