Logarithm: Definition, Rules, Properties, Formulas, Examples

A logarithm is used to express any power of a number. In this section, we will learn about logarithms with examples and properties.

Table of Contents

Definition of Logarithm

For $a>0, a\neq 1$ and $M>0,$ assume that $a^x=M.$ In this case, the number $x$ is said to be the logarithm of $M$ with respect to the base $a,$ and it is written as

\[x=\log_a M\]

This can be 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 $a^x=M.$ Thus we have

\[a^x=M \iff x=\log_a M \]

Few Examples of Logarithms

Ex 1. Firstly we provide an example of the logarithm of a whole number. Note that $3^2=9.$ In terms of the logarithm, this power rule can be expressed as \[2=\log_3 9\] Therefore, we can say that $3^2=9 \iff 2=\log_3 9$

Ex 2. Next, we give an example of the logarithm of a fraction. It is known that $10^{-2}=0.01.$ This power rule can be expressed in terms of the logarithm as \[-2=\log_{10} 0.01\]. Hence, from the definition of the logarithm we obtain the following relation: $10^{-2}=0.01 \iff -2=\log_{10} 0.01$

In a similar way, we can get more examples:

Ex 3. $2^3=8 \iff 3=\log_2 8$

Ex 4. $5^0=1 \iff 0=\log_5 1$

Ex 5. $7^{-1}=\frac{1}{7} \iff -1=\log_7 \frac{1}{7}$ ♣

Some Remarks on Logarithms

(R1) The logarithm of a number does not make any sense if we do not mention the base.

(R2) Recall from the definition of the logarithm that $a^x=M \iff x=\log_a M$. Here, if $a^x$ (or $M$) is a negative number then the value of $x$ will be an imaginary complex number. Thus, the logarithm of a negative number is imaginary.

(R3) The logarithm of any number $a$ with respect to the base $a$ is $1.$ In other words, \[\log_a a=1.\]

Proof: Note that $a^1=a.$ Thus, by the definition of logarithms, we obtain that $\log_a a=1.$

(R4) The logarithm of $1$ with respect to any base $a (a \neq 0)$ is $1.$ In other words, \[\log_a 1=0.\]

Proof: We know that $a^0=1$ for any $a \ne 0.$ Then by the definition of logarithms, we have $\log_a 1=0.$ ♣

Main Properties of Logarithms

The formulas/rules of logarithms are given below:

• Product Rule of Logarithm: $\log_a(MN)=\log_a M +\log_a N$

• Quotient Rule of Logarithm: $\log_a(M/N)=\log_a M -\log_a N$

• Power Rule of Logarithm: $\log_a M^k=k\log_a M$

• Base Change Rule of Logarithm: $\log_a M=\log_b M \cdot \log_a b$

Types of Logarithms:

Natural Logarithm: The logarithm of a number with respect to the base $e$ is called the natural logarithm. From the definition, it is clear that the natural logarithm is denoted by $\log_e$. We represent this logarithm as $\ln$. Thus, we have $\log_e=\ln.$

Example: The natural logarithm of $e$ is $\ln(e)=\log_e e=1$ as $\log_a a=1.$

Common Logarithm: The logarithm of a number with respect to the base $10$ is called the common logarithm. So the common logarithm is denoted by $\log_{10}.$

Example:The common logarithm of $10$ is $\log_{10} 10=1$ as $\log_a a=1.$

Remark: If the base of a logarithm is not mentioned, then that logarithm is calculated with base $10.$

Solved Examples on Logarithms

Ex 1: Find $\log_2 8$

Solution:

Note that $8=2 \times 2 \times 2=2^3$

∴ $\log_2 8=\log_2 2^3=3 \log_2 2$ [by the power rule of logarithms]

$=2 \cdot 1$ [$\because \log_a a=1$]

$=2$

Ex 2: Find $\log_3 \sqrt{27}$

Solution:

We know that $27=3 \times 3 \times 3=3^3$

∴ $\sqrt{27}=(3^3)^{1/2}=(3)^{3 \times 1/2}=3^{3/2}$

It follows that $\sqrt{27}=3^{3/2}$

Now, $\log_3 \sqrt{27}=\log_3 (3)^{3/2}$ $=\frac{3}{2} \log_2 2$ [by the power rule of logarithms]

$=\frac{3}{2} \cdot 1$ [$\because \log_a a=1$]

$=\frac{3}{2}$

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