# exponentials and logarithms

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

- Created by: Nicola
- Created on: 24-06-10 01:19

First 294 words of the document:

1

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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