Sunday, 20 July 2008

Logarithm | The Common Mistakes Created

Making mistakes is interesting. Sounds funny?

We LEARN through mistakes. And Logarithm enables learners to "create" those common mistakes (unknowingly).

First of all, let me explain the syntax of writing the Logarithmic term.

Logarithmic term is expressed as logaY, where the symbol "a" is known as the "base".

NOTE: If the base is 10, normally we will leave the logarithmic term as logY (without writing the base 10). The explanation below will use base 10 for simplicity.

The study of Loagrithm involves 3 powerful logarithmic laws.

With these laws, any logarithmic expressions can be easily simplified. Here are the 3 laws:

1) Product Law
  • log (XY) = log X + logY ==> the log terms are adding

2) Quotient Law
  • log(X / Y) = log X - log Y ==> the log terms are subtracting

3) Power Law
  • logXn = n log X ==> the power n is brought in front of the term.

Common mistakes made:
  • Writing log X + log Y as log (X + Y) ==> they are not equal

  • Thinking that "log" and "X " are separated ==> they are together "logX "

  • Writing log (X/Y) as log X / log Y ==> It is "X divided by Y" before being "log".

Example of application of the Laws:

Simplify log x2 + logy - log (xy)

Step 1: Identify the laws that can be used ==> Both Product & Quotient Laws are OK.
Step 2: Since the first two terms are added, we apply Product Law ==> log[x2y]
Step 3: As the last term is subtracted we use Quotient Law ==> log[x2y / (xy)]

step 3: The "(xy)" is taken as a group and becomes the denominator as a whole. This is because log (xy) means operating "log" onto (xy), not " log x" times "y".

The result simplifies to log x (answer). Is logarithm it simple?

A little tip ==> log nn = 1. The log of a number with the same base equals ONE!
This is useful if we are to combine a number with a logarithmic expression. See below example.

Simplify ( log X ) - 1.

Solution: The "1" can be converted to the Logarithm "log 1010" or simply "log 10". The working therefore becomes (log X) - log10, which results in log (X/10).

Why did we convert "1" to "log with base 10" ? It is because the first term log X is of base 10. Therefore to be able to combine both terms, we must select the "1" to convert to the same base as the first term "log X".

Make sense?

In summary, Logarithm is simple. Be aware of the writing form of Logarithm, and understand them.

Do not fear mistakes. We can learn from these mistakes, but, after looking and understanding these mistakes, we should correct and not make them again. OK?


Anonymous said...

thank you very much teacher, i have been looking for all these complex explanation and it is you that make maths more interesting.

Please keep it up your hard work.

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