# Logarithm – Definition, Rules , Properties and Examples

A Logarithm is a mathematical function that represents the exponent to which a fixed number, known as the base, must be raised to produce a given number. In other words, it is the inverse operation of exponentiation.

Logarithms are widely used in science, engineering, and mathematics for dealing with very large or very small numbers, as they allow for easier manipulation and comparison of exponential rates of growth or decay. They are foundational in fields such as acoustics, electronics, and in the analysis of algorithms, where they help in understanding the complexity and performance.

Let’s learn **logarithms in detail, including logarithmic functions, Logarithm rules, Logarithm properties, Logarithm graphs, and Logarithm examples.**

Table of Content

- What is Logarithm?
- Definition of Logarithm | Log Definition
- Logarithm Types
- Difference between Log and ln | log vs ln
- Rules of Logarithm | Log Rules
- Product Rule of Log
- Quotient Rule of Log
- Power Rule of Log
- Change of Base Rule
- Base Switch Rule
- Equality of Logarithm
- Number Raised to Log
- Negative Log Rule
- Articles related to Logarithms:
- Log 1
- Logaa
- Log 0
- Logarithmic Function
- Expanding and Condensing Logarithm
- Logarithmic Formulas
- Log Calculator
- Log Table
- Anti Log Table
- Logarithmic Graph
- Properties of Logarithmic Graph
- Solved Examples on Logarithm
- Practice Questions on Logarithm

## What are Logarithms?

If a^{n} = b then log or logarithm is defined as the log of b at base a is equal to n. It should be noted that in both cases base is ‘a’ but in the log, the base is with the result and not the power.

a^{n}= b ⇒ log_{a}b = n

where,

is Base**a**is Argument**b**and**a**Positive Real Numbers**b**is Real Number**n**

### Logarithms Meaning

Logarithms are a mathematical concept used to represent the exponent or power to which a base number must be raised to produce a given number.

### Exponential to Log Form

If a number is expressed in the exponential form for example an = b where a is the base, n is the exponent and b is the result of the exponent then to convert it into the logarithmic form the base ‘a’ remains base in logarithm, the result ‘b’ becomes an argument and the exponent ‘n’ becomes the result here.

a^{n}= b ⇒ log_{a}b = n

### Log to Exponential Form

If the expression is in logarithmic form then we can convert it into exponential form by making the argument as a result and the result of logarithm becomes the exponent while the base remains the same. It can be better understood from the expression mentioned below:

log_{a}b = n ⇒ a^{n}= b

## Definition of Logarithm | Log Definition

A logarithm is a mathematical concept that answers the question: to what exponent must a given base number be raised to produce a specific number? In simpler terms, if you have an equation of the form *b** ^{y}* =

*, then the logarithm of*

*x**to base*

*x**is*

*b**, expressed as y = log*

*y*_{b}(x).

## Logarithm Types

Depending upon the base, there are two types of logarithm, which are,

- Common Logarithm
- Natural Logarithm

### Common Logarithm

The logarithm with base 10 is known as ** Common Logarithm**. It is written as log

_{10}X. The common logarithm is generally written as

**only instead of**

**log**

**log****.**

_{10}Let’s see some examples.

- log10 = log
_{10}10 = 1 - log1 = log
_{10}1 = 0 - log1000 = log
_{10}1000 = 3

### Natural Logarithm

The logarithm with base e, where e is a mathematical constant is called ** Natural Logarithm**. It is written as log

_{e}X. The natural logarithm is also written in the abbreviated form as

**i.e. log**

**ln**_{e}X = ln X.

Let’s see some example:

- log
_{e}2 = X ⇒ e^{X}= 2 - log
_{e}5 = y ⇒ e^{y}= 5

## Difference between Log and ln | log vs ln

The basic difference between log and ln is tabulated below:

Log | Ln |
---|---|

Log is the logarithmic expression at base 10. | Ln is the logarithmic expression at base e. |

It is given as log_{10}X | It is given as log_{e}x |

It is called a common log and is represented as log x | It is called natural log and is represented as ln x |

It is mostly used for solving large numbers and simplifying calculations. | It is less commonly used |

**Learn More: ****Difference Between Log and Ln**

## Rules of Logarithm | Log Rules

The common properties or rules of Log are :

- Product Rule
- Division Rule
- Power Rule
- Change of Base Rule
- Base Switch Rule
- Equality of Log
- Number raised to Log Power
- Negative Log Rule

** Read in Detail:** Logarithm Rules | List of all the Log Rules with Examples

### Product Rule of Log

Product rule of log states that if log is applied to the product of two numbers then it is equal to the sum of the individual logarithmic values of the numbers. The expression can be given as:

log_{x}ab = log_{x}a + log_{x}b

** Example:** loga10 = loga(5 ✕ 2) = loga5 + loga2

### Quotient Rule of Log

Quotient Rule of log states that if log is applied to the quotient of two numbers then it is equal to the difference of the individual logarithmic value of the numbers. The expression can be given as:

log_{x}a/b = log_{x}a – log_{x}b

** Example:** log

_{x}2 = log

_{x}(10/5) = log

_{x}10 – log

_{x}5

### Power Rule of Log

Power Rule of log states that if the argument is raised to some power then the solution of logarithmic expression is given by the power of the argument multiplied by the log value of the argument. The expression can be given as:

log_{x}a^{b}= b.log_{x}a

### Change of Base Rule

In logarithm, the base can be changed in the following way

log_{x}a = log_{y}a/log_{y}b

log_{x}a.log_{y}b = log_{y}a

### Base Switch Rule

This log property states that Base and argument can be switched in the following manner

log_{x}a = 1/log_{a}x

### Equality of Logarithm

Equality of logarithm property states that if

log_{x}a = log_{x}b then a = b

### Number Raised to Log

If a number is raised to log which has the same base as the number then the result of the expression is the argument. This can be expressed as

[Tex]x^{log_{x}a}~=~a[/Tex]

### Negative Log Rule

The Negative Log Property states that if the logarithmic expression is of the form -log_{x}a then we can convert it into a positive form by taking the reciprocal of the argument or by taking the reciprocal of the base as

-log_{x}a = log_{x}(1/a) = log_{1/x}a

**Articles related to **Logarithms:

**Articles related to**

Apart from the above-mentioned properties, there are some other properties of Log. Using these properties we can directly put their values in any equation. These properties are mentioned below:

### Log 1

This property of log states that the value of Log 1 is always zero, no matter what the base is. This is because any number raised to power zero is 1. Hence, Log 1 = 0.

### Log_{a}a

This Property of log states that if the base and augment of a logarithm are the same then the logarithm of that number is 1. This is because any number raised to power 1 results in the number itself. Hence,

- ln e = log
_{e}e = 1

- log
_{10}10 = 1

- log
_{2}2 = 1

### Log 0

This rule states that the log of zero is not defined as there is no such number when raised to any power that results in zero. Hence, log 0 = Not defined.

## Logarithmic Function

A logarithmic function is the inverse of an exponential function and is defined for positive real numbers with a positive base (not equal to 1). The logarithmic function to the base * b* is represented as f(x) = log

_{b}(x), where x>0 and

*>0. In this function, X is the argument of the logarithm, and b is the base.*

*b*## Expanding and Condensing Logarithm

A logarithmic expression can be expanded and condensed using the following Properties of Log:

: log**Product Rule of Log**_{x}ab = log_{x}a + log_{x}b: log**Quotient Rule of Log**_{x}a/b = log_{x}a – log_{x}b: log**Power Rule of Log**_{x}ab = b.log_{x}a

### Expanding Log

Log can be expanded in the following manner.

**Example: Expand log(2a**^{3}**b**^{2}**)**

**Solution:**

log(2a

^{3}b^{2}) = log 2 + log a^{3}+ log b^{2}= log 2 + 3 log a + 2 log b

### Condensing Log

A log can be condensed in the following manner just by following the reverse of the properties of log.

**Example: Condense log 2 + 3 log a + 2 log b**

**Solution:**

log 2 + 3 log a + 2 log b

= log 2 + log a3 + log b2

= 2a

^{3}b^{2}

## Logarithmic Formulas

The formulas for logarithm is tabulated below:

Logarithmic Formulas | |
---|---|

log1 | 0 |

log_{x}x | 1 |

log_{x}(ab) | log_{x}a + log_{x}b |

log_{x}(a/b) | = log_{x}a – log_{x}b |

log_{x}(a)^{b} | = b.log_{x}a |

log_{x}a | = log_{y}a/log_{y}b |

log_{x}a | = 1/log_{a}x |

-log_{x}a | log_{x}(1/a) = log_{1/x}a |

** Learn More :** Logarithm Formulas

## Log Calculator

Try out the following calculator to find the value of log of different numbers at various bases

## Log Table

Log Table is used to find the value of the log without the use of a calculator. The log table provides the logarithmic value of a number at a particular base.

A log table has mainly three columns. The first column contains two-digit numbers from 10 to 99, the second column contains differences for digits 0 to 9 and hence called the difference column and the third column contains mean difference from 1 to 9 and hence called ** mean difference column**.

The log table for base ** e** is called the natural logarithm table and that for base 2 is called the binary log table.

The logarithmic value of a number contains two parts named ** characteristics** and

**both separated by a decimal. Characteristic is the integral part written o the left side of the table and can be positive or negative while the mantissa is the fraction or decimal that is always positive.**

**mantissa**** Read More :** Log Table

## Anti Log Table

Antilog is the process of finding the inverse of the log of the number. This is used when the number is already given in log value and we need to find out the number for which log value is given. If log a = b then a = antilog (b).

Antilog table is helpful in finding the Antilog value without using the calculator. The antilog table also consists of 3 columns among which the first column contains numbers from .00 to .99, the second block which is the difference column contains digits from 0 to 9, and the third block which is the mean difference column contains digits from 1 to 9.

Using Log Table we have find out the characteristics and mantissa of a number while using the antilog table we will separate the characteristics and mantissa of the number.

** Read More:** Antilog Table

## Logarithmic Graph

We know that the domain of Logarithmic Function is (0, ∞) and its range is a set of all real numbers. If we plot the graph using the set of domain and range we find that the graph of the logarithmic function is just the inverse of the graph obtained for the exponential function.

This indicates the inverse relationship between exponential and logarithmic functions. Also, the logarithmic graph is symmetric around the line y = x. We know that the value of log 1 is zero at any base value. Hence it has an intercept (1,0) on the x-axis and no intercept on the y-axis as log 0 is not defined.

## Properties of Logarithmic Graph

There are the following properties of the logarithmic Graph of function log_{a}x

- In logarithmic function base a > 0 and a ≠ 1
- The graph of logarithmic function increases when a > 1 and decreases in the range 0 < a < 1.
- The domain of the function is a set of all positive numbers greater than zero.
- The range of the curve is a set of all real numbers.

## Logarithms Applications

Logarithms are extensively utilized in science and technology today. Logarithmic calculators, which simplify complex calculations, are readily available. They are applied in areas like surveying, celestial navigation, measuring sound levels (decibels), assessing earthquake intensity (Richter scale), analyzing radioactive decay, and determining acidity (pH = -log10[H+]).

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## Solved Logarithms Examples

**Example 1: Find log**_{a}**16 + 1/2 log**_{a}**225 – 2log**_{a}**2**

**Solution:**

log

_{a}16 + 1/2 ✕ 2log_{a}15 – log_{a}2^{2}⇒ log

_{a}16 + log_{a}15 – log_{a}4⇒ log

_{a}(16 ✕ 15) – log_{a}4⇒ log

_{a}(16 ✕ 15/4) = log_{a}60

**Example 2: Solve log**_{b}**3 – log**_{b}**27**

**Solution:**

log

_{2}3 – log_{2}48⇒ log

_{2}(3/48)⇒ log

_{2}(1/16)⇒ log

_{2}(-16)⇒ -log

_{2}2^{4}⇒ -4log

_{2}2 = -4

**Example 3: Find x in log**_{b}**x + log**_{b}**(x – 3) = log**_{b}**10**

**Solution:**

Given log

_{b}x + log_{b}(x – 3) = log_{b}10⇒ log

_{b}(x)(x – 3) = log_{b}10⇒ (x)(x – 3) = 10

⇒ x

^{2 }– 3x – 10 = 0⇒ x

^{2}– 5x + 2x – 10 = 0⇒ x(x – 5) + 2(x – 5)

⇒ (x – 5)(x + 2) = 0

⇒ x = 5, -2

## Practice Questions on Logarithm

Practice logarithms with solved examples to increase your understanding on the topic:

** Q1.** Solve log

_{2}8

** Q2.** Show that log

_{e}e = 1 where is euler’s number

** Q3.** Find y if log

_{3}(y-5) = 1

** Q4.** Find the value of log 2 using log table.

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## Logarithm – FAQs

### What are Logarithms?

Logarithm is the inverse of exponential which is used to find out to what power a base must be raised to yield a particular value.

### What are Different Types of Logarithms?

There are two types of Logarithms:

- Common Logarithm (Base 10)
- Natural logarithm (Base e)

### Who Invented Logarithm?

Logarithm was invented by Scottish Mathematician John Napier in 1614.

### Can Logarithm be Negative?

No the argument of logarithm can’t be negative, however, log of any number can give a negative value.

### What are Logarithms used for?

Logarithm is used to find the power a number must be raised to yield a particular result. It is useful in finding pH level, exponential growth or decay, etc.

### What is the value of log_{a}0?

Logarithm does not take 0 and negative numbers as its input hence log

_{a}0 is not defined.