# Introduction to Logarithm

## Logarithms: Definition, Examples, and Properties

In this section, we will learn about logarithms with examples and properties.

### Definition of Logarithm:

We consider a>0, ane 1 and M>0,  and assume that
a^x=M.
In this case, we will call x to be the logarithm of M with respect to the base a.
We write this phenomenon as
x=log_a M
(Read as: “x is the logarithm of M to the base a”)
therefore a^x=M Rightarrow x=log_a M
On the other hand, if x=log_a M then we havea^x=M.
To summarise, we can say that
a^x=M if and only if x=log_a M.
We now understand the above definition with examples.

### Examples of Logarithm:

1).  We know that 2^3=8.
In terms of logarithms, we can express it as
3=log_2 8
therefore 2^3=8 iff 3=log_2 8
2).  Note that 10^{-1}=frac{1}{10}=0.1
That is, 10^{-1}=0.1
According to the logarithms, we have
-1=log_{10} 0.1
Thus, 10^{-1}=0.1 iff -1=log_{10} 0.1

### Remarks on Logarithms:

(A) If we do not mention the base, then there is no meaning of the logarithms of a number.
(B) The logarithm of a negative number is imaginary.
(C) log_a a=1.
Proof:  As a^1=a, the proof follows from the definition of the logarithm.
(D)  log_a 1=0.
Proof:   For any a ne 0, we have a^0=1. Now applying the definition of logarithms, we obtain the result.

### Properties of Logarithms:

Logarithm has the following four main properties
a). log_a(MN)=log_a M + log_a N
This is called the product rule of logarithms.
b). log_a(M/N)=log_a M – log_a N
This is called the Quotient Rule of Logarithms
c). log_a M^n=n log_a M
This is called the Power Rule of Logarithms
d). log_a M=log_b M times log_a b
This is the Base Change Rule of Logarithms

### Solved Examples:

Ex1:  Find log_3 27
Note that we have 27=3^3
So by the definition of the logarithm, we have
log_3 27=3 ans.

Ex2:  Find log_2 sqrt{8}
We have 8=2^3
therefore sqrt{8}=(2^3)^{1/2}=(2)^{3 times 1/2}=2^{3/2}
Thus, sqrt{8}=2^{3/2}
Now, log_2 sqrt{8}=log_2 (2)^{3/2}=3/2 log_2 2=3/2 ans.
(by the above power rule of logarithms and log_a a=1)