Introduction to Logarithmic Functions

Before the development of calculus, many mathematicians used logarithms to convert multiplication and division into addition and subtraction problems.

The logarithmic function is an important element for mathematical computations. Scottish mathematician, physicist, and astronomer John Napier made the first formal study of logarithms in the 16th century.

Numerous astronomical and scientific calculations requiring large numbers can use it.

The exponential function is considered the inverse of the logarithmic function because of their close relationship.

The logarithmic function  $$log_{a}{N}= x$$ is created from the exponential function   $$a^{x}= N$$

What is Logarithm Function?

The simplest definition is logarithm is “what is the power to which a number must be raised to get some other values.”? A logarithmic function in mathematics is the opposite of exponentiation. In other words, logarithms are a different way of thinking about exponents.

Let’s say we want to know 2 raised to the power of 2nd power equals 9. The exponential equation expresses this is 3

We know that 2 raised to the 4th power equals 16; we can also express this with an exponential equation $$2^4=16$$

Now, imagine we are asked, “2 raised to which power is equal to 16?” The answer is 4. We can express this by the following logarithmic equation: $$log_{2}{16}= 4$$ we can read as “log base two of sixteen equal four.”

Introduction To Functions Properties –

Both equations represent the exact relationship between the numbers 2, 4, and 16,


  • 2 represents the base
  • And 4 represents the exponent.

The table below contains more examples of equivalent logarithmic and exponential equations.

Introduction To Functions Properties –

The definition of the logarithmic function is

If If $$x > 0$$and $$a > 0$$ and $$a ≠ 1$$

If and only if $$x = a^{x}$$ then $$y= log_{a}{x}$$

The function is therefore defined as f(x) = $$log_{a}{x}$$

The logarithm’s base is a. You can interpret this as log base a of x.

The logarithmic functions base 10 and base e are the two most prevalent bases used in logarithmic operations.

Key Differences Between Common Logarithm (Log) and Natural Logarithm (Ln)

The table below contains the main differences between common logarithm (Log) and natural logarithm (Ln).

Log: Common LogarithmLn: Natural Logarithm
Log means logarithm to the base 10Ln means logarithm to the base e
Log is generally referred to as a “common logarithm.”Ln is known as a “natural logarithm.

f(x) = $$log_{10}{x}$$
f(x) = $$log_{e}{x}$$
The exponent form of the common logarithm is 10x =yThe exponent form of the natural logarithm is ex =y
The question for the common logarithm is, “At which number should one raise 10 to get y?”The question for the natural logarithm is, “At which number should one raise Euler’s constant number to get y?”
Common logarithm functions are more widely used in chemistry and physics than ln. Natural Logarithms are used in solving time and growth problems.
– In advanced mathematics, the natural logarithm is the logarithm of choice. Several unique properties of the natural logarithm functions and their inverse functions make life much easier in calculus.
The common logarithmic function has a base of 10. The natural logarithmic function, also known as the logarithmic function to the base e.
The table Displays The Main Differences Between Common Logarithm (Log) and Natural Logarithm (Ln) –

Properties of Logarithm

A single logarithm can be expanded into many logarithms or compressed into many logarithms by using the properties of a log function. Just another approach to writing exponents is with a logarithm. As a result, the characteristics of exponents are used to infer the characteristics of logarithms.

The rules of logarithms, which are derived from the exponent rules, are what make up the attributes of a log function.

These logarithm features are used to simplify logarithmic expressions and solve logarithmic problems.

The Image Displays The Properties Of Logarithm Functions –

Where x and y > 0 and n is a real number.

Properties of Exponents

Exponent-based expressions can be made simpler by using the laws of exponents or properties. The principles of exponents make it simple and rapid to execute numerous arithmetic operations like addition, subtraction, multiplication, and division.

These guidelines are also useful for reducing the complexity of numbers that have complex powers incorporating fractions, decimals, and roots.

The process of simplifying exponent-based equations is facilitated by exponent rules, commonly referred to as the “laws of exponents” or the “properties of exponents.”

These guidelines can be used to help simplify expressions that contain exponents that are decimals, fractions, irrational numbers, or negative integers.

Let x and y be real numbers, and a and b be integers. Below are some of the properties of exponents, including that all of the expressions appearing in equations are defined.

Image Displays The Properties of Exponents –

Exponential Functions

A mathematical function called an exponential function is employed frequently in everyday life. Exponential functions are mostly used to calculate investments, model populations, determine exponential decline or exponential growth, and so forth.

The formula for an exponential function is $$f(x) = e^x$$, where x is a variable, and an is a constant that serves as the function’s base and must be bigger than 0. The transcendental number e, or roughly 2.71828, is the most often used exponential function basis.

Image Contains Exponential Functions’ Properties –

Where x, y, and n are real numbers.

Comparison of Exponential and Logarithmic Functions

The table below compares the properties of exponential and logarithmic functions.

Comparison of Exponential and Logarithmic Functions
Table Contains Comparison of Exponential and Logarithmic Functions –

Logarithm Functions in Real Time

  • When radiocarbon dating, logarithms are utilized to measure radioactive decay.
  • In particular, in calculations, where multiplications are converted to additions and logarithms are employed.
  • To estimate the exponential population growth, logarithms are also used.
  • Calculus also involves logarithmic calculations. In the actual world, these computations are applied to several calculations.
  • Problems in combinatorics are solved using logarithms. A particular area of mathematics called combinatorics focuses on investigating finite discrete structures.


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