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LEARNING AND UNDERSTANDING LOGARITHM


In mathematics many ideas are related.
We saw that addition and subtraction are related and that multiplication and division are related.
Similarly, exponentials and logarithms are related. Logarithms, commonly referred to as logs, are the inverse of exponentials. The logarithm of a number x in the base a is defined as the number n such that an=x. So, if a to the power n = x, then: log to the power a (x) = n .

DEFINITION OF LOGARITHM

The logarithm of a number is the value to which the base must be raised to give that number i.e. the exponent. From the first example of the activity log to the power 2 (4) means the power of 2 that will give 4.
As 2 to the power 2 = 4, we see that log to the power 2 (4) = 2 The exponential-form is then 2 to the power 2 = 4 and the logarithmic-form is log to the power 2 4=2.

Definition 1: Logarithms If a to the power n = x, then: log to the power a (x) = n, where a > 0; a ≠ 1 and x > 0.


Activity 1: Logarithm symbols Write the following out in words. The first one is done for you.

1. log to the power 2 (4) is log to the base 2 of 4

2. log to the power 10 (14)

3. log to the power 16 (4)

4. log to the power x (8)

5. log to the power y (x)


Activity 2: Applying the definition Find the value of:
1. log to the power 7 343 Reasoning: 7 to the power 3 = 343 therefore, log to the power 7 343 = 3

2. log to the power 2 8


3. log to the power 4 1/64

4. log to the power 10 1000.

Logarithm Bases


Logarithms, like exponentials, also have a base and log to the power 2 (2) is not the same as log to the power 10 (2).

We generally use the ‘common’ base, 10, or the natural base, e.

The number e is an irrational number between 2,71 and 2,72. It comes up surprisingly often in Mathematics, but for now suffice it to say that it is one of the two common bases.

Extension — Natural logarithm:


The natural logarithm (symbol ln) is widely used in the sciences. The natural logarithm is to the base e which is approximately 2,718 281 83. e, like π, is an example of an irrational number. While the notation log to the power 10 (x) and log to the power e (x) may be used, log to the power 10 (x) is often written log (x) in Science and log to the power e (x) is normally written as ln (x) in both Science and Mathematics. So, if you see the log symbol without a base, it means log to the power 10.
It is often necessary or convenient to convert a log from one base to another.
An engineer might need an approximate solution to a log in a base for which he does not have a table or calculator function, or it may be algebraically convenient to have two logs in the same base.

Logarithms can be changed from one base to another, by using the change of base formula: log to the power a x = log to the power b x/ log to the power b a where b is any base you find convenient. Normally a and b are known, therefore log to the power b a is normally a known, if irrational, number.

For example, change log to the power 2 12 in base 10 is: log to the power 2 12 = log to the power 10/log to the power 10 2 .
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