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 log

_{a}Y, 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**

- logX
^{n }= 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 x

_{2 }+ 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[x

_{2}y]

Step 3: As the last term is subtracted we use Quotient Law ==> log[x

_{2}y / (xy)]

**NOTE**

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

_{n}n = 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

_{10}10" or simply "log 10". The working therefore becomes (log X) - log10, which results in

**log (X/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".*

**Why did we convert "1" to "log with base 10" ?**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?

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