Logarithm Formula

Logarithm Formula

Sep 08, 2022, 16:45 IST

In mathematics, logarithms are another way of writing exponents. The logarithm of a number with a base is equal to other numbers. The logarithm is just the opposite function of exponentiation. For e.g, if 10 ² = 100, then log ₁₀ 100 = 2.

Log _b x = n or b ⁿ = x

Where b = base of the logarithmic function.

This can be noticed as the “Logarithm of x to the base b is equal to n.” In this short article, we will learn the definition of logarithms, two types of logarithms such as common logarithm and natural logarithm, and various properties of logarithms with many examples.

What is a logarithm?

The logarithm is a power to which a number must be raised to obtain additional values. It is the most convenient way of expressing large numbers. The logarithm has various important properties that prove that multiplication and division of logarithms can also be written in the logarithm form of subtraction and addition.

A logarithm of a positive real number "a" with respect to the base b, a positive real number is not equal to 1[ ^{nb 1} ], is the exponent by which b must be raised to give a.

i.e. b ^y = a ⇔log _b a=y

Where

• a and b = two positive real numbers
• y = real number
• a = argument, which is inside the log
• b = base, which is at the bottom of the log

Below are some examples of conversion from exponential forms to logarithms.

History

John Napier introduced the concept of Logarithms within the 17th century. Later it had been employed by many scientists, navigators, engineers, etc for performing various calculations which made it simple. In simple words, It is the inverse process of the exponentiation. During this article, we are getting to have a glance at the definition, properties, and samples of logarithm intimately.

Types of Logarithm

It is of two types which is mention below:

1. Common Logarithm
2. Natural Logarithm

Common Logarithm

The common logarithm is also known as the base 10 logarithms. It is represented as log10 or simply log. For example, the common logarithm of 1000 is written as a log (1000). The common logarithm defines how many times we have to multiply the number 10, to get the required output.

For example, log ₁₀ (100) = 2

If we multiply the number 10 twice, we get the result 100.

Natural Logarithm

The natural logarithm is called the base e logarithm. The natural logarithm is represented as ln or loge. Here, “e” represents the Euler’s constant which is approximately equal to 2.71828. For example, the natural logarithm of 78 is written as ln 78. The natural logarithm defines how many we have to multiply “e” to get the required output.

For example, ln (78) = 4.357

Thus, the base e logarithm of 78 is equal to 4.357.

Logarithms Rules and Properties

We have provided below certain rules based on which logarithmic operations can be performed:

1. Product Rule

Log _b (mn)= log _b m + log _b n

For example: log ₃ ( 4y ) = log ₃ (4) + log ₃ (y)

2. Division Rule

Log _b (m/n)= log _b m – log _b n

For example, log ₃ ( 5/ y ) = log ₃ (5) -log ₃ (y)

3. Exponential Rule

Log _b (m ⁿ ) = n log _b m

Example: log _b (4 ³ ) = 3 log _b 4

4. Change of Base Rule

Log _b m = log _a m/ log _a b

Example: log _b 3 = log _a 3/log _a b

5. Base Switch Rule

log _b (a) = 1 / log _a (b)

Example: log _b 7 = 1/log ₇ b

6. Derivative of log

If f (x) = log _b (x), then the derivative of f(x) is given by;

f'(x) = 1/(x ln(b))

Example: Given, f (x) = log ₈ (x)

Then, f'(x) = 1/(x ln(8))

7. Integral of Log

∫log _b (x)dx = x( log _b (x) – 1/ln(b) ) + C

Example: ∫ log ₈ (x) dx = x ∙ ( log ₈ (x) – 1 / ln(8) ) + C

Other Properties

Some related properties of logarithmic functions are:

• Log _b b = 1
• Log _b 1 = 0
• Log _b 0 = undefined

Basic Logarithm formulas

• log _b (mn) = log _b (m) + log _b (n)
• log _b (m/n) = log _b (m) – log _b (n)
• Log _b (xy) = y log _b (x)
• Log _b m√n = log _b n/m
• m log _b (x) + n log _b (y) = log _b (x ^m y ⁿ )
• log _b (m+n) = log _b m + log _b (1+nm)
• log _b (m – n) = log _b m + log _b (1-n/m)

Solved Examples

Q1. Solve log ₂ (32) = ?

Ans. Since 2 ⁶ = 2 × 2 × 2 × 2 × 2 = 32, 5 is the exponent value and log ₂ (32)= 5.

Q2. What is the value of log ₁₀ (100)?

Ans. log ₁₀ (100)= 2

Q3. Use of the property of logarithms, solve for the value of x for log ₃ x= log ₃ 5+ log ₃ 4

Ans. By the addition rule, log ₃ 5+ log ₃ 4= log ₃ (5 * 4 )

Log ₃ ( 20 ). Thus, x= 20.

Q4. S olve for x in log ₂ x = 4

Ans. This logarithmic function can be written In the exponential form as 2 ⁴ = x

Therefore, 2 ⁴ = 2 × 2 × 2 × 2 = 16, X = 16.

Frequently Asked Question (FAQs)

Q1. What are the three properties of logarithms?

Ans. The three properties of logarithm are:

1. Product rule: am. an=a. m+n
2. Quotient rule: am/an = a. m-n
3. Power rule: (am)n = a. mn

Q2. What is domain of log?

Ans. The domain of the log function y = log x is x > 0 or (0, ∞). The range of any log function is set of all real numbers (R)

Q3. What is the natural log of 0?

Ans. Undefined

Q4. What is the opposite of log?

Ans. As we know that the inverse of a log function is an exponential. So, we know that the inverse of f(x) = log subb(x) is f ^-1 (y) = b ^y .

Q5. How will you write 1 as a log?

Ans. A logarithm of x =1 is the number y we have to raise the base b to get the value 1. So, the base 10 logarithm of 1 is 0.