## 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 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:

**Find `log_3 27`**

__Ex1:__Note that we have `27=3^3`

So by the definition of the logarithm, we have

`log_3 27=3`

__ans.__

**Find `log_2 sqrt{8}`**

__Ex2:__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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