Logarithm Formula

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, b^x = 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 logarithm is stated below,

• log_a(mn)  = log_am + log_an (Product property)

Quotient Formula of Logarithms

Quotient Formula of logarithm is stated below,

• log_a(m/n) = log_am – log_an (Quotient property)

Power Formula of Logarithms

Power Formula of logarithm is stated below,

• log_a(mⁿ) = nlog_am (Power property)

Change of Base Formula

Base of the a Lograthin is changed using the formula,

• log_ba =  (log_ca)/(log_cb) (Change of Base Property)

Other Logarithm Formulas

Various others Logarithm Formulas are,

• log_b(a√n) = 1/a log_bn
• log of 1 = log_a1 = 0
• log_aa = 1 (Identity rule)
• log_ba= log_bc =>  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 = log_ex. 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.

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• Natural Log Formula

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(x^y) = 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,

log_b (xy) = log_b x + log_b y

This is derived as,

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

Now using definition of logarithm,

x = b^m and y = bⁿ

x.y = b^m × bⁿ = b^{(m + n)} → {by a law of exponents, p^m × pⁿ = p^{(m + n)}}

x.y = b^{(m + n)}

Converting into logarithm form,

m + n = log_b xy

from eq. (1)

log_b (xy) = log_b x + log_b y

Derivation of Quotient Formula of Log

Quotient formula of log states that,

log_b (x/y) = log_b x – log_b y

This is derived as,

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

Now using definition of logarithm,

x = b^m and y = bⁿ

x/y = b^m / bⁿ = b^{(m – n)} → {by a law of exponents, a^m / aⁿ = a^{(m – n)}}

Converting into logarithm form

m – n = logb (x/y)

from eq. (1)

log_b (x/y) = log_b x – log_b y

Derivation of Power Formula of Log

Power formula of log state that,

log_b a^x = x log_b a

This is derived as,

Let log_b a = m….(i)

Now using definition of logarithm,

a^x = (b^m)^x

a^x = (b^mx) {by a law of exponents, (a^m)ⁿ = a^mn}

Converting into logarithm form,

log_b a^x = m x

using eq. (i)

log_b a^x = x log_b a

Derivation of Change of Base Formula of Log

Change of base formula of log states that,

log_b a = (log_c a) / (log_c b)

This is derived as,

Let, log_b a = x, log_c a = y, and log_c b = z

In exponential forms,

a = b^x … (1)

a = c^y … (2)

b = c^z … (3)

From (1) and (2),

b^x = c^y

(c^z)^x = c^y (from (3))

c^zx = c^y

zx = y

x = y / z

Substituting values of x, y, and z back,

log_b a = (log_c a) / (log_c 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.

Read More

• Negative Exponents
• How to multiply and divide exponents
• Adding and Subtracting of  Exponents
• Laws of Logarithims

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

2^x = 8

2^x = 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 = ±√5 [Log of Negative Number is Not Defined]

x = √5

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

Solution:

Given log₉ (59049) [9⁵= 59049]

= log₉(9)⁵

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

= 5

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

Solution:

Given log₁₀(5) + 1

= log₁₀(5) + log₁₀10 [Identity Rule]

= log₁₀(5 × 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 = ±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?

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

2.How To Derive Log Formulas?

Logarithm 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 Logarithm?

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

5. What is Quotient Formula of Logarithm?

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

6. What is Power Formula of Logarithm?

The power formula of logarithm states that, for any base ‘n’, log_n(a)^b = b.log_n(a)