The concept of Logarithms was introduced by John Napier in the \(17th\) century. Then, later, many scientists, navigators, engineers, etc., made it easy for performing various calculations. So, logarithms are the opposite of exponentiation. There are several rules known as the laws of logarithms.

These allow expressions involving logarithms to be rewritten in a variety of ways. The regulations apply to logarithms of any base, but the same floor must be used throughout a calculation. In this article, we will learn more about logarithms, their laws and proofs.

Logarithm Definition

Definition: The logarithm is defined using the exponent as follows.

\({b^x} = a \Leftrightarrow {\log _b}a = x\)

Here, the right side of the arrow is read to be a logarithm of \(a\) to the base \(b\) is equal to \(x\).

Here,

1. \(a\) and \(b\) are considered as the two positive real numbers.
2. \(x\) is the real number.
3. \(a\) which is inside the \(log\), is known as the argument.
4. \(b\), which is at the bottom of the \(log\), is known as the base.

Examples: If \({6^2} = 36\) and the logarithm will be \({\log _6}36 = 2\)

Laws of Logarithm Definition

Definition: The logarithmic expressions can be written in various ways, and there are a few specific laws called the laws of logarithms.

That is \({b^v} = a\), which is expressed as \({\log _b}a = y\).

In other words, this can be stated as the logarithm of a positive real number \(a\) to the base \(b\), is a positive \(y\).

Examples: 

\({10^1} = 10 \Rightarrow 10 = 1\)
\({10^2} = 100 \Rightarrow 100 = 2\)
\({10^3} = 1000 \Rightarrow 1000 = 3\)
\({10^4} = 10000 \Rightarrow 10000 = 4\)
\({3^2} = 9 \Rightarrow 9 = 2\)
\({3^5} = 243 \Rightarrow 243 = 5\)
and so on.

Types of Logarithms

You can see the two types of logarithm in most of the cases, and they are:

1. Common Logarithm
2. Natural Logarithm

1. Common Logarithm

The common logarithm is also known as the base ten logarithms. It is written as \(p\log \log p\). So, when the logarithm is taken with respect to base \(10\), then we call it is the common logarithm.

The base is sometimes not written in a common logarithm. In some logarithms, if the base is not written, we assume that it is a common logarithm with base \(10\).

Examples: 
\({10^1} = 10 \Rightarrow 10 = 1\) or simply we write\(\log \log 10 = 1\)
\({10^2} = 100 \Rightarrow 100 = 2\)
\({10^3} = 1000 \Rightarrow 1000 = 3\)
\({10^4} = 10000 \Rightarrow 10000 = 4\)

2. Natural Logarithm

Examples:

\({2.71828^1} = 2.71828 \Rightarrow {e^1} = 2.71828 \Rightarrow 2.71828 = 1\) , or, \(e=1\) or simply \(ln e=1\)
\({2.71828^2} = 7.39 \Rightarrow {e^2} = 7.39 \Rightarrow 7.39 = 2\) , or, \(7.39=1\) or simply ln \(7.39=2\)
\({2.71828^3} = 20.08 \Rightarrow {e^3} = 20.08 \Rightarrow 3\), or, \(20.08=3\) or simply \(ln 20.08=3\)
and, so on.

Rules and Properties of Logarithm

We have specific rules based on which the logarithmic operations can be achieved, which are given below:

1. Product rule
2. Division rule
3. Power rule or the Exponential Rule
4. Change of base rule
5. Base switch rule
6. Derivative of log
7. Integral of log

Now, we will discuss this one by one along with the examples:

1. Product rule

The base remains the same, the sum of the logarithms of two numbers is equal to the product of the logarithms of the numbers.

It is written as \(\log a + \log b = \log ab\)

Example: (a) \({\log _2}5 + {\log _2}4 = {\log _2}(5 \times 4) = {\log _2}20\)
(b) \({\log _{10}}6 + {\log _{10}}3 = {\log _{10}}(6 \times 3) = {\log _{10}}18\)
(c) \(\log \log x + \log \log y = \log \log (x \times y) = \log \log xy\)
(d) \(\log 4x + \log \log x = \log \log (4x \times x) = \log \log 4{x^2}\)

The natural logarithm is known as the base \(e\) logarithm, where \(e\) is the Euler’s constant, which is approximately equal to \(2.71828\). 
The natural logarithm is written as \(ln\) or \(x\) . 

2. Division rule

The base remains the same, the logarithm of the quotient of two numbers is equal to the difference of the logarithms of those two numbers.
\({\log _b}\left( {\frac{m}{n}} \right) = {\log _b}m – {\log _b}n\)

Example: \({\log _3}\left( {\frac{2}{y}} \right) = {\log _3}(2) – {\log _3}(y)\)

3. Power rule

In this rule, the base remaining the same, the logarithm of \(m\) to a rational exponent, is equal to the exponent times the logarithm of \(m\).
\({\log _b}\left( {{m^n}} \right) = n{\log _b}m\)

Example: \({\log _b}\left( {{2^3}} \right) = 3{\log _b}2\)

4. Change of base rule: Sometimes, in mathematical calculations involving logarithm, we need to change the base of the logarithm. This rule allows a change of base of the logarithm.

Suppose we have the logarithm of a number \(m\) with respect to the base \(b\). We want to change the base \(b\) to the base \(a\). In this case, we can change the base as follows:

\({\log _b}m = \frac{{{{\log }_c}m}}{{{{\log }_a}b}}\)

Example: \({\log _b}2 = \frac{{{{\log }_a}2}}{{{{\log }_a}b}}\)

5. Base switch rule

\({\log _b}(a) = \frac{1}{{{{\log }_e}(b)}}\)

Example: \({\log _b}8 = \frac{1}{{{{\log }_8}b}}\)

6. Derivative of Log

If \((x) = (x)\)  then the derivative of \(f(x)\) is given as:
\(f(x) = 1/(x\ln \ln (b))\)

Example: Given, \(f(x) = (x)\)

Then, \(f(x) = 1/(x\ln \ln (10))\)

7. Integral of Log

\(\int {(x)} dx = x((x) – 1/\ln \ln (b)) + c\)

Example: \(\int {(x)} dx = x((x) – 1/\ln \ln (10)) + c\)

Some Other Properties

Few more properties of the logarithm functions are given below:

1. \({\log _b}b = 1\)
2. \({\log _b}1 = 0\)
3. \({\log _b}0 = \)
4. The logarithm of a negative number is not defined

How to Use the Logarithm Properties?

We use the Logarithm properties to make the calculations simpler:

Logarithms help us to convert the exponential form \({2^5} = 32\) into logarithmic form \({\log _2}32 = 5\)
The logarithm property is helpful to write the product as the sum.

\(\log \log 14 = \log \log (7 \times 2) = \log 7 + \log 2\)
The logarithm property is utilised to write the division as the difference.

\(\log 0.3 = \log \log \frac{3}{{10}} = \log \log 3 – \log 10\)
The logarithm property is utilised to write the exponent as the product

\(\log \sqrt 5 = \log {5^{\frac{1}{2}}} = \frac{1}{2}\log 5\)
The logarithm property is utilised to split the large number into smaller factors.

\(\log \log 24 = \log \log (8 \times 3) = \log \log \left( {{2^3} \times 3} \right) = \log {2^3} + \log \log 3 = 3\log 2 + \log 3\)

Formulas of Logarithm

There are a few of the formulas of logarithm given below:

1. \({\log _b}({\rm{mn}}) = {\log _b}(\;{\rm{m}}) + {\log _b}(n)\)
2. \({\log _b}\left( {\frac{m}{n}} \right) = {\log _b}(m) – {\log _b}(n)\)
3. \({\log _b}\left( {{x^y}} \right) = y{\log _b}(x)\)
4. \({{\mathop{\rm mlog}\nolimits} _b}(x) + {{\mathop{\rm nlog}\nolimits} _b}(y) = {\log _b}\left( {{x^m}{y^n}} \right)\)
5. \({\log _b}(m + n) = {\log _b}m + {\log _b}\left( {1 + \frac{n}{m}} \right)\)
6. \({\log _b}(m – n) = {\log _b}m + {\log _b}\left( {1 – \frac{n}{m}} \right)\)
7. \({\log _b}b = 1\)
8. \({\log _b}1 = 0\)
9. \({\log _b}m = \frac{{{{\log }_a}m}}{{{{\log }_a}b}}\)

Solved Examples – Laws of Logarithm

Q.1. Evaluate the given expression: \({\log _2}8 + {\log _2}4\).
Ans: Here, we will apply the rule,
\({\log _2}8 + {\log _2}4 = {\log _2}(8 \times 4)\)
\( = {\log _2}32\)
Now, rewrite \(32\) in the exponential form to get the value of its exponent.
\(32 = {2^5}\)
So, we have, \({\log _2}32 = {\log _2}{2^5} = 5{\log _2}2 = 5 \times 1 = 5\)
Hence, the required answer is \(5\).

Q.2. Evaluate [{\log _3}162 – {\log _3}2].
Ans: Here, you have to apply the quotient rule law.
\({\log _3}162 – {\log _3}2 = {\log _3}\left( {\frac{{162}}{2}} \right) = {\log _3}81\)
Now, write the argument in the exponential form,
\(81 = {3^4}\)
So, we have, \({\log _3}81 = {\log _3}{3^4} = 4{\log _3}3 = 4 \times 1 = 4\)
Hence, the required answer is \(4\).

Q.3. Find the value of \(x\) giving, \(2\log \log x = 4\log 3\)
Ans: Given \(2\log x = 4\log 3\)
Here, you have to divide each side by the number \(2\).
\( \Rightarrow \log \log x = \frac{{4\log 3}}{2}\)
\( \Rightarrow \log \log x = 2\log 3\)
\( \Rightarrow \log \log x = \log {3^2}\)
\( \Rightarrow \log \log x = \log 9\)
Hence, \(x = 9\)

Q.4. Calculate \(\log \log x + \log \log (x – 1) = \log (3x + 12)\)
Ans: Given,\(\log \log x + \log \log (x – 1) = \log (3x + 12)\)
\( \Rightarrow \log \log [x(x – 1)] = \log (3x + 12)\)
Now, you have to drop the logarithms to get:
\( \Rightarrow [x(x – 1)] = (3x + 12)\)
Here, you have to use the distributive property to remove the brackets.
\(\Rightarrow {x^2} – x = 3x + 12\)
\( \Rightarrow {x^2} – x – 3x – 12 = 0\)
\( \Rightarrow {x^2} – 4x – 12 = 0\)
\( \Rightarrow (x – 6)(x + 2) = 0\)
\( \Rightarrow x = – 2,x = 6\)
The logarithm is not defined for negative real numbers. Hence, the required answer is \(x = 6\)

Q.5. Evaluate the given, \({\log _2}(5x + 6) = 5\)
Ans: Given, \({\log _2}(5x + 6) = 5\)
Now, rewrite the given equation in the exponential form,
\({2^5} = 5x + 6\)
Simplify,\(32 = 5x + 6\)
Here, subtract 6 from both sides of the equation.
\(32 – 6 = 5x + 6 – 6\)
\( \Rightarrow 26 = 5x\)
\( \Rightarrow x = \frac{{26}}{5}\)
Hence, the required answer is given above.

Summary

In the given article, we have discussed what the logarithm of a number is, provided examples, and gave the definition for logarithm. We have concerned the rules and the properties of the logarithms with an example for each rule and property. Finally, we glanced at the types of common and natural logarithms—then provided a few formulas used in logarithms.

We have also talked about a few points on how to use the properties of the logarithm. The use and application of logarithm help us deal the large numbers in a very easy way.

FAQs

Q.1. How many types of logarithms are there?
Ans: There are two types of logarithms, and they are given below:
Common Logarithm
The common logarithm is also known as the base ten logarithms. It is written as \(p\log \log p\) . So, when the logarithm is taken with respect to base \(10\), then we call it is the common logarithm.
Example: 
\({10^2} = 100 \Rightarrow 100 = 2\)
Natural Logarithm
The natural logarithm is known as the base \(e\) logarithm, where \(e\) is the Euler’s constant, which is approximately equal to \(2.71828\). 
The natural logarithm is written as ln or \(x\) . 
Example:
\({2.71828^4} = 54.6 \Rightarrow {e^4} = 54.6,{\rm{or}},54.6 = 4\) or simply ln \(56.6=4\)
And so on.

Q.2. Explain Laws of Logarithm with example?
Ans: The laws of the logarithms are given below:
1. Product rule
The base remains the same, the sum of the logarithms of two numbers is equal to the product of the logarithms of the numbers.
It is written as \(\log a + \log b = \log ab\)
Example: \({\log _2}5 + {\log _2}4 = {\log _2}(5 \times 4) = {\log _2}20\)
2. Division rule
The base remaining the same, the logarithm of the quotient of two numbers is equal to the difference of the logarithms of those two numbers.
\({\log _b}\left( {\frac{m}{n}} \right) = {\log _b}m – {\log _b}n\)
Example: \({\log _3}\left( {\frac{2}{y}} \right) = {\log _3}(2) – {\log _3}(y)\)
3. Power rule or Exponential rule
In this rule, the base remaining the same, the logarithm of m to a rational exponent, is equal to the exponent times the logarithm of \(m\).
\({\log _b}\left( {{m^n}} \right) = n{\log _b}m\)
Example: \({\log _b}\left( {{2^3}} \right) = 3{\log _b}2\)
4. Change of base rule
Suppose we have the logarithm of a number \(m\) with respect to the base \(b\). We want to change the base \(b\) to the base \(a\). In this case, we can change the base as follows:
\({\log _b}m = \frac{{{{\log }_c}m}}{{{{\log }_e}b}}\)
Example: \({\log _b}2 = \frac{{{{\log }_s}2}}{{{{\log }_a}b}}\)

Q.3. What does \(3\log \) mean?
Ans: The \(3\log \) reduction is defined as reducing a large number by \(1000\) times. 

Q.4. What is a log divided by a log?
Ans: When you divide two exponential numbers with the same base \(b\), we subtract their exponents. 
In the case of division, the base remains the same, the logarithm of the quotient of two numbers is equal to the difference of the logarithms of those two numbers.
\({\log _b}\left( {\frac{m}{n}} \right) = {\log _b}m – {\log _b}n\)
Example: \({\log _7}\left( {\frac{3}{p}} \right) = {\log _7}(3) – {\log _7}(p)\)

Q.5. What is the \(\log \) law?
Ans: The logarithm is defined using the exponent.
\({b^x} = a \Leftrightarrow {\log _b}a = x\)
Here, the right side of the arrow is read to be a logarithm of \(a\) to the base \(b\) is equal to \(x\).
Here,
1. \(a\) and \(b\) are considered as the two positive real numbers.
2. \(x\) is the real number.
3. \(a\) which is inside the log, is known as the argument.
4. \(b\), which is at the bottom of the log, is known as the base.
Examples: \({6^2} = 36\) is written in terms of the logarithm as \({\log _6}36 = 2\)

We hope this detailed article on the full log table helps you. If you have any queries regarding this article, reach out to us through the comment section below and we will get back to you as soon as possible.