Logarithms | Definition, Concepts, and Rules

By Alex David

April 15, 2025

In mathematics, there is another way of looking differently at the exponent. It is known as logarithm.

History of Logarithms

A logarithm is a combination of two Greek words: “logos” and “arithmos” which can be translated as “ratio” or “proportion.” John Napier invented this word: “logarithms.” Its invention was based on the comparison of arithmetic and geometric sequences.

Logarithms is a mathematical operator. Logarithm is another way to think about an exponent. In fact, it is an inverse operation of exponentiation. It answers the question, “To what exponent must the base be raised to produce a given number?”

Definition of Logarithms

“A logarithm is the power to which a given base must be raised to obtain a specific number.”

Any positive real number R can be expressed in two possible ways: the first is in exponential form and the second is in logarithmic form. For example, R = 36, this positive real number can be written in exponential form as: 36 = $6^2$ , and the same number can be written as $\log_6 36 = 2$ .

Mathematically, if

$b^x = y$

then in the logarithm form, it is written as

$\log_b(y) = x$

where:

b is the base (must be greater than 0 and not equal to 1)
y is the number we are taking the logarithm of
x is the exponent.

What is the Logarithm Function?

The logarithm function is a mathematical function that represents the logarithm of a number with a given base.

Mathematical Form

$f(x) = \log_b(x)$

where:

b is the base (b > 0 and b ≠ 1)
x is the input (must be positive)
f(x) gives the exponent to which the base must be raised to produce x

Properties

The domain is (0,∞) (only positive numbers)
The range is (−∞,∞)
The function passes through (1,0) because $\log_b(1) = 0$ for any base b
It is an increasing function if b > 1 and a decreasing function if 0 < b < 1

Types of Logarithms

There are different types that are divided on the basis of their base and usage.

Common Logarithms (Base 10)
Natural Logarithms (Base e)
Binary Logarithms (Base 2)
Negative Logarithms

Common Logarithms

“A common logarithm is a type of logarithm that is defined as the logarithm function with base 10.”

It is also known as Briggsian logarithm. It has another name, which is decimal logarithm.

Mathematical Form

If:

$10^y = x$

then the common logarithm is:

$\log(x) = y$

Example

$\log(10) = 1$ because $10^1 = 10$
$\log(100) = 2$ because $10^2 = 100$

Properties

It is used in the decimal number system.
It is used in the fields of engineering, physics, and chemistry.
It is used for logarithmic scales

Natural Logarithms

A natural logarithm is defined as the log function with base ‘e’, where ‘e’ is Euler’s number. The ‘e’ is a constant value and it is approximately equal to 2.71828.

Mathematical Form

$e^y = x$

then the natural logarithm

$\ln(x) = y$

Example

$\ln(e) = 1$ because $e^1 = e$
$\ln(1) = 0$ because $e^0 = 1$

Properties

Base e is an irrational number.
It is used in calculus, exponential growth and decay, and financial modeling.

Binary Logarithms

It is a logarithm with base 2.

Mathematical Form

$\log_2(x) = y$

Example

$\log_2(8) = 3$

Properties

It is widely used in digital systems.
It is used in data compression and complexity analysis.
It is used to determine the number of bits required to represent a number in binary form.

Negative Logarithms

What happens if the input number is between 0 and 1? It is about negative logarithms. It occurs when the logarithm of a number is negative.

Mathematical Form

$\log_b(x) = y$

where 0 < x < 1

Example

$\log_{10}(0.1) = -1$

Properties

It is used to measure pH values in chemistry and signal processing.

What are characteristic and mantissa?

Common logarithm can be solved with the help of characteristic and mantissa. Logarithm of positive numbers consist of two parts:

Characteristic: The integer part of the logarithm.
Mantissa: The decimal part of the logarithm, always positive.

Application of Logarithms

It is used to find the magnitude of earthquakes with the help of the Richter scale.
It is used to determine the intensity of sound with the help of the decibel scale.
Logarithmic functions are used to find the compound interest, exponential growth, and decay.
It is used to find the pH level of the substance.
It is used to determine radioactive decay.

Conclusion

Logarithm is the inverse operation of the exponent. It has vast usage in modern technologies. Types of logarithm help to understand all the concepts and make it easy for learning. There are properties of logarithms and identities of logarithm that will help the most in calculating logarithms.

Frequently Asking Questions (FAQs)

Q # 01: Why do logarithm exist, and what problem do they solve?

Logarithms simplify complex multiplications and divisions into simple additions and subtractions, making large calculations easier in science, engineering, and computing.

Q # 02: What is the difference between logarithm and exponents?

Exponents show repeated multiplication, while logarithms find which exponent was used to reach a number.

Example

$10^3 = 1000$ (exponent form)

$\log_{10}(1000) = 3$ (logarithmic form)

Q # 03: Why can’t logarithm have a base of 1 or negative numbers?

A logarithm with base 1 is meaningless because any number raised to any power still equals 1. Logarithms with negative bases are undefined in real numbers because they create complex values.

Q # 04: How are logarithm used in computing and data science?

Logarithms are used in:

Machine learning algorithms

Example: Feature scaling in logistic regression.

Big data processing

Example: Logarithmic complexity in algorithms like binary search.

Compression techniques

Example: Reducing data size using log functions.

Q # 05: How do logarithm apply to probability and statistics?

Logarithms are used in entropy calculations, likelihood estimation, and regression models, helping to process large numerical datasets efficiently.

Q # 06: What is the difference between discrete and continuous logarithm?

Discrete logarithms are used in cryptography and number theory.
Continuous logarithms appear in calculus and real-world modeling (e.g., exponential decay).

Q # 07: Why do logarithmic scales help in measuring natural phenomena?

Logarithmic scales compress large ranges of values, making them ideal for earthquake magnitudes (Richter scale), sound intensity (decibels), and pH levels in chemistry.