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Logarithms – Definition, Rules , Properties and Examples

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Logarithm is a mathematical function 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.

Let’s learn logarithms in detail, including logarithmic functions, their curve, and examples.

History of Logarithm

The creation of logarithms is credited to a Scottish Mathematician John Napier. In 1614 he presented his book named Canons of Logarithms which contained a table of trigonometric functions and their natural logarithms. The purpose of the book was to help in the multiplication of quantities.

Logarithm Definition

If an = 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.

an = b ⇒ logab = n

where, 

  • a is Base
  • b is Argument
  • a and b Positive Real Numbers
  • n is Real 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.

an = b ⇒ logab = 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:

logab = n ⇒ an = b

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 log10X. The common logarithm is generally written as log only instead of log10.

Let’s see some examples.

  • log10 = log1010 = 1
  • log1 = log101 = 0
  • log1000 = log101000 = 3

Natural Logarithm

The logarithm with base e, where e is a mathematical constant is called Natural Logarithm. It is written as logeX. The natural logarithm is also written in the abbreviated form as ln i.e. logeX = ln X.

Let’s see some example:

  • loge2 = X ⇒ eX = 2
  • loge5 = y ⇒ ey = 5

Difference between Log and ln

The basic difference between log and ln is tabulated below:

LogLn
Log is the logarithmic expression at base 10.Ln is the logarithmic expression at base e.
It is given as log10XIt is given as logex
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

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Rules of Logarithm

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

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:

logxab = logxa + logxb

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:

logxa/b = logxa – logxb

Example: logx2 = logx(10/5) = logx10 – logx5

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:

logxab = b.logxa

Change of Base Rule

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

logxa = logya/logyb

logxa.logyb = logya

Base Switch Rule

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

logxa = 1/logax

Equality of Logarithm

Equality of logarithm property states that if

logxa = logxb 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 -logxa 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

-logxa = logx(1/a) = log1/xa

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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.

Logaa

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 = logee = 1
  • log1010 = 1
  • log22 = 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.

Expanding and Condensing Logarithm

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

  • Product Rule of Log: logxab = logxa + logxb
  • Quotient Rule of Log: logxa/b = logxa – logxb
  • Power Rule of Log: logxab = b.logxa

Expanding Log

Log can be expanded in the following manner.

Example: Expand log(2a3b2)

Solution:

log(2a3b2) = log 2 + log a3 + log b2

= 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

= 2a3b2

Logarithmic Formulas

The formulas for logarithm is tabulated below:

Logarithmic Formulas

log10
logxx1
logx(ab)logxa + logxb
logx(a/b)= logxa – logxb
logx(a)b= b.logxa
logxa= logya/logyb
logxa= 1/logax
-logxalogx(1/a) = log1/xa

Learn More :

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 mantissa 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.

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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.

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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.

Logarithmic Graph

Logarithmic Graph

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 logax

  • 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.

Related :

Logarithm Examples

Example 1: Find loga16 + 1/2 loga225 – 2loga2

Solution:

loga16 + 1/2 ✕ 2loga15 – loga22

⇒ loga16 + loga15 – loga4

⇒ loga(16 ✕ 15) – loga4

⇒ loga(16 ✕ 15/4) = loga60

Example 2: Solve logb3 – logb27

Solution:

log23 – log248

⇒ log2(3/48)

⇒ log2(1/16)

⇒ log2(-16)

⇒ -log224

⇒ -4log22 = -4

Example 3: Find x in logbx + logb(x – 3) = logb10

Solution:

Given logbx + logb(x – 3) = logb10

⇒ logb(x)(x – 3) = logb10

⇒ (x)(x – 3) = 10

⇒ x2 – 3x – 10 = 0

⇒ x2 – 5x + 2x – 10 = 0

⇒ x(x – 5) + 2(x – 5)

⇒ (x – 5)(x + 2) = 0

⇒ x = 5, -2

Logarithm Practice Questions

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

Q1. Solve log28

Q2. Show that logee = 1 where is euler’s number

Q3. Find y if log3(y-5) = 1

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

Frequently Asked Questions on Logarithms

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 loga0?

Logarithm does not take 0 and negative numbers as its input hence loga0 is not defined.



Last Updated : 16 Feb, 2024
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