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Logarithms are another way of writing indices.

You may often see and written, with no base indicated. It is generally recognised that this is shorthand.

Contents 1 Laws of logs 1.1 Example 1.2 Typical Examination Question (OCR MEI) 2 Comments

Laws of logs

The properties of indices can be used to show that the following rules for logarithms hold:

Example

Simplify:

NB: In the above example, I have not written what base each of the logarithms is to. This is because for the laws of logarithms, it doesn't matter what the base is, as long as all of the logs are to the same base.

Another important law of logs is as follows. This is a very useful way of changing the base (in this formula, the base does matter!). Most calculators can only work out ln x and lg x (usually just written as 'log' on the button) so this formula can be very useful.

Typical Examination Question (OCR MEI)

The example below covers a logarithm problem in the same format as many Logarithm Section B problems covered by the OCR MEI specification. (The question below comes from the C2 Specimen Paper (2004))

A virus is spreading through a population and so a vaccination programme is introduced

Thereafter, the number of new cases are as follows

Week Number, x ¦ 1 ¦ 2 ¦ 3 ¦ 4 ¦ 5 ¦ Number of New cases, y ¦ 240 ¦ 150 ¦ 95 ¦ 58 ¦ 38 ¦

The number of new cases,y, in week x is to be modelled by an equation of the form y = pq^x where p and q are constants.

Question 1 - Show how the equation above can be mapped into a linear equation for finding the value of p and q.

If we consider the expression y = pq^x and apply logarithms according to laws of logarithms above, we can see how this forms a linear expression:

log y = log pq^x --> Apply Logs log y = log p + log q^x --> Consider the log rule that equates multiplication to addition. log y = logp + xlog q --> Consider the rule for indices, where a power is transposed to the front of the expression.

(More coming soon!)

Comments

I don't find this a very useful article yet. It doesn't state the laws of logs (even though it uses them and mentions them). It also does not explain what a log really is and what all the bases they talk about are. It is not suitable for someone who has trouble and wants to try and learn the topics from a fresh perspective, nor does it go in to enough details, nor advanced work for someone wishing to get a greater or more advanced understanding of the wider topic.

(Previous to addition of example)