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 
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 have`a^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`)
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