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

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 $a$”)

∴ a^{x} =M ⇒ 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**

**Examples of Logarithm**

**1).** We know that 2^{3} =8.

In terms of logarithms, we can express it as

3 = log_{2}8

∴ 2^{3} = 8 ⇔ 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 ⇔ -1 = log_{10} 0.1

**Remarks of Logarithm**

**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) **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 ≠ 0, we have a^{0} =1. Now applying the definition of logarithms, we obtain the result.

**Properties of Logarithm**

**Properties of Logarithm**

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 × log_{a} b.

This is the Base Change Rule of logarithms

**Solved Examples**

**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)

**ALSO READ**

**FAQs**

**FAQs**

**Q1: What are logarithms?**

Answer: Logarithms are used to express exponents in other ways. More specifically, the exponent a^{x} =M in terms of the logarithm can be expressed as x=log_{a} M.

**Q2: What is the logarithm of 1?**

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