Definition of logarithms

Introduction

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 an=x, then:         

loga(x)=n

Definition of logarithms

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 log2(4) means the power of 2 that will give 4. As 22=4, we see that

log2(4)=2

(1)

The exponential-form is then 22=4 and the logarithmic-form is log24=2.

Definition 1: Logarithms

If an=x, then: loga(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. log2(4) is log to the base 2 of 4
2. log10(14)
3. log16(4)
   
   
   4. logx(8)
   
   
   5   logy(x)

Activity 2: Applying the definition

Find the value of:

1. log7343
   Reasoning:73=343therefore,log7343=3

   2    log28
   
   3  log4164
   
   4  log101000



Logarithm bases



Logarithms, like exponentials, also have a base and log2(2) is not the same as log10(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 log10(x) and loge(x) may be used, log10(x) is often written log(x) in Science and loge(x) is normally written as ln(x) in both Science and Mathematics. So, if you see the log symbol without a base, it means log10.
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:
logax=logbxlogba
(1)
where b is any base you find convenient. Normally a and b are known, therefore logba is normally a known, if irrational, number.
For example, change log212 in base 10 is:
log212=log1012log102


Activity 1: Change of base
Change the following to the indicated base:
1. log2(4) to base 8
2. log10(14) to base 2
3. log16(4) to base 10
4. logx(8) to base y
5. logy(x) to base x                                                                                                                      o base x