Logarithms: Definition, Examples, and Properties

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

Definition of Logarithm 

We consider $a>0, a \ne 1$ and $M>0$,  and assume that

ax =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= loga M

(Read as: “$x$ is the logarithm of $M$ to the base $a$”)

∴ ax =M  ⇒  x=loga M

On the other hand, if x=loga M then we have ax =M.

To summarise, we can say that

ax =M if and only if x=loga M.

We now understand the above definition with examples.

Examples of Logarithm

1).  We know that 23 =8.

In terms of logarithms, we can express it as

3 = log28

∴ 23 = 8 ⇔ 3 = log28

2).  Note that $10^{-1}=\frac{1}{10}=0.1$

That is, 10-1 = 0.1

According to the logarithms, we have

-1 = log10 0.1

Thus, 10-1 = 0.1 ⇔ -1 = log10 0.1

Remarks of Logarithm

(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) loga a=1.

Proof:  As a1 =a, the proof follows from the definition of the logarithm.

(D)  loga 1=0.

Proof:   For any a ≠ 0, we have a0 =1. Now applying the definition of logarithms, we obtain the result.

Properties of Logarithm

Logarithm has the following four main properties

a). loga(MN) = logaM + logaN

This is called the product rule of logarithms.

b). loga(M/N) = logaM – logaN

This is called the Quotient Rule of logarithms

c). logaMn =n loga M

This is called the Power Rule of logarithms

d). loga M = logb M × loga b.

This is the Base Change Rule of logarithms

Solved Examples

Ex1:  Find log327

Note that we have 27=33.

So by the definition of the logarithm, we have

log3 27=3 ans.

Ex2:  Find $\log_2 \sqrt{8}$

We have 8=23

$\therefore \sqrt{8}=(2^3)^{1/2}=(2)^{3 \times 1/2}=2^{3/2}$

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 loga a=1)



Q1: What are logarithms?

Answer: Logarithms are used to express exponents in other ways. More specifically, the exponent ax =M in terms of the logarithm can be expressed as x=loga M.

Q2: What is the logarithm of 1?

Answer: The logarithm of 1 with any base is always 0.

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