# Logarithm Formula

Last Updated : 02 Jan, 2024

Logarithm was invented in the 17th century by Scottish mathematician John Napier (1550-1617). The Napier logarithm was the first to be published in 1614. Henry Briggs introduced a common (base 10) logarithm. John Napier’s purpose was to assist in the multiplication of quantities that were called sines.

## Logarithm Formula

A logarithm is defined as the power to which a number is raised to yield some other values. Logarithms are the inverse of exponents. There is a unique way of reading the logarithm expression. For example, bx = n is called as ‘x is the logarithm of n to the base b. There are two parts of the logarithm: Characteristic and Mantissa. The integral part of a logarithm is called ‘Characteristic’ and the decimal part which is non-negative is called ‘Mantissa’. The characteristic can be negative but mantissa can’t. For example log10(120) = 2.078 ( 2 is characteristic and .078 is mantissa).

## Properties of Logarithm

Logarithmic Expressions follow different properties. The different properties of logarithms are mentioned below:

### Product Formula of Logarithms

Product Formula of logarithim is stated below,

• loga(mn)  = logam + logan (Product property)

### Quotient Formula of Logarithms

Quotient Formula of logarithim is stated below,

• loga(m/n) = logam – logan (Quotient property)

### Power Formula of Logarithms

Power Formula of logarithim is stated below,

• loga(mn) = nlogam   (Power property)

### Change of Base Formula

Base of the a Lograthin is changed using the formula,

• logba =  (logca)/(logcb) (Change of Base Property)

### Other Logaritim Formulas

Various others Lagarithim Formulas are,

• logb(aâˆšn) = 1/a logbn
• log of 1 = loga1 = 0
• logaa = 1 (Identity rule)
• logba= logbc =>  a= c (Equality rule)
• = x (Raised to log)

## Natural log

The natural logarithm of a number is its logarithm to the base ‘e’. ‘e’ is the transcendental and irrational number whose value is approximately equal to 2.71828182. It is written as ln x. ln x = logex. It is a special type of logarithm, used for solving time and growth problems. It is also used for solving the equation in which the unknown appears as the exponent of some other quantity.

## Properties of Natural Log

Properties of Natural Log are,

### Product Rule

The product rule of natural log states that,

ln(xy)  = ln(x) + ln(y)

### Quotient Rule

The quotient rule of natural log states that,

ln(x/y) = ln(x)  – ln(y)

### Reciprocal Rule

The reciprocal rule of natural log states that,

ln(1/x) = -ln(x)

### Log of Power

The log of any term that is written in power term is written as,

ln(xy) = y ln(x)

### Natural Log of e

The natural log of “e” is always 1(one) as the base in natural log is ‘e’. This is represented as,

ln (e) = 1

### Log of 1

The log of 1 is always zero.

ln (1) = 0

## Log Formulas Derivation

Log formulas are very useful for solving various mathematical problems and these formula are easily derived using laws of exponents. Now lets learn about the derivation of some log formulas in detail.

### Derivation of Product Formula of Log

Product formula of log states that,

logb (xy) = logb x + logb y

This is derived as,

Let take, logb x = m and logb y = n…(i)

Now using definition of logarithm,

x = bm and y = bn

x.y = bm Ã— bn = b(m + n) â†’ {by a law of exponents, pm Ã— pn = p(m + n)}

x.y = b(m + n)

Converting into lograthim form,

m + n = logb xy

from eq. (1)

logb (xy) = logb x + logb y

### Derivation of Quotient Formula of Log

Quotient formula of log states that,

logb (x/y) = logb x – logb y

This is derived as,

Let take, logb x = m and logb y = n…(i)

Now using definition of logarithm,

x = bm and y = bn

x/y = bm / bn = b(m – n) â†’ {by a law of exponents, am / an = a(m – n)}

Converting into lograthim form

m – n = logb (x/y)

from eq. (1)

logb (x/y) = logb x – logb y

### Derivation of Power Formula of Log

Power formula of log state that,

logb ax = x logb a

This is derived as,

Let logb a = m….(i)

Now using definition of logarithm,

ax = (bm)x

ax = (bmx) {by a law of exponents, (am)n = amn}

Converting into lograthim form,

logb ax = m x

using eq. (i)

logb ax = x logb a

### Derivation of Change of Base Formula of Log

Change of base formula of log states that,

logb a = (logc a) / (logc b)

This is derived as,

Let, logb a = x, logc a = y, and logc b = z

In exponential forms,

a = bx … (1)

a = cy … (2)

b = cz … (3)

From (1) and (2),

bx = cy

(cz)x = cy (from (3))

czx = cy

zx = y

x = y / z

Substituting values of x, y, and z back,

logb a = (logc a) / (logc b)

## Applications of Logarithm

Various applications of Logarithm are,

• Logarithm is Used for expressing larger value.
• Logarithm is Used for measuring earthquake intensity.
• Logarithm is Used for measuring pH value.
• Logarithm is Used for modeling business applications
• Logarithm is Used by scientists to determine the rate of radioactive decay
• Logarithm is Used by economists for plotting the graphs.

## Solved Examples on Logarithm Formula

Example 1: Solve log2(x) = 4

Solution:

log2(x) = 4

24 = x

x = 16

Example 2: Solve log2(8) = x

Solution:

log2(8) = x

2x = 8

2x = 23

x = 3

Example 3: Find the value of x if log6(x – 3) = 1.

Solution:

log6(x – 3) = 1

61 = (x – 3)

x – 3 = 6

x = 9

Example 4: Find  x if log(x – 2) + log(x + 2) = log21

Solution:

log(x – 2) + log(x + 2) = log21

log(x – 2) + log(x + 2) = 0 [log(1) =0]

log[(x – 2)(x + 2)] = 0 [Product Rule]

(x – 2)(x + 2) = 1 [Antilog(0) = 1]

x2 – 4 = 1

x2 = 5

x = Â±âˆš5 [Log of Negative Number is Not Defined]

x = âˆš5

Example 5: Find the value of log9(59049).

Solution:

Given log9 (59049) [95= 59049]

= log9(9)5

= 5.log9(9) (identity rule i.e logaa]

= 5

Example 6: Express log10(5) + 1 in form of log10x

Solution:

Given log10(5) + 1

= log10(5) + log1010 [Identity Rule]

= log10(5 Ã— 10) [Product Rule]

= log1050

Example 7: Find the value of x if log10(x2 – 15) = 1.

Solution:

log10(x2 – 15) = 1

log10(x2 – 15) = log1010 [Identity Rule]

Applying Antilog,

(x2 – 15) = 10

x2 = 25

x = Â±5

## Practice Questions on Logarithm Formula

Q1. Find the value of x: 3.log(x) = log 27

Q2. Simplify log2(16) + 2.log3(9)

Q3. Find the value of x: 2.log(2x) = log 81

Q4. Simplify log3(9) – 3.log3(27)

Q5. Simplify ln(x3/y2z)

## FAQs on Logarithm Formula

### 1. What are Logarithm Formulas?

Logarithim Formulas are the formulas that are useful to solve the logarithimic problems. They are derived using laws of exponents.

### 2.How To Derive Log Formulas?

Logarithim Formulas are derived using Laws of Exponetnts

### 3.What are Applications of Log Formulas?

Various applications of log formulas are,

• They are used to simplify log problems.
• They are used to solve various large calculation.
• They are used to find the Derivative and Integral of various functions.
• They are used in graph plotting, etc.

### 4. What is Product Formula of Logarithim?

The product formula of logarithim states that, for any base ‘n’, logn(a.b) = logn(a) + logn(b)

### 5. What is Quotient Formula of Logarithim?

The quotient formula of logarithim states that, for any base ‘n’, logn(a/b) = logn(a) – logn(b)

### 6. What is Power Formula of Logarthim?

The power formula of logarithim states that, for any base ‘n’, logn(a)b = b.logn(a)

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