Logarithmic Functions

Introduction

Many mathematicians have utilised logarithms to convert multiplication and division problems into addition and subtraction problems before the discovery of calculus. The power of some numbers (typically, the base numbers) is raised to obtain another number in logarithms. It is the reciprocal of the exponential function.

The inverses of exponential functions are logarithmic functions, and any exponential function can be represented in logarithmic form. All logarithmic functions can be rewritten in exponential form in the same way. Logarithms are extremely important because they allow us to work with very large quantities while altering numbers that are much smaller.

Here, we’ll go through the definition and formula for the logarithmic function, logarithmic function properties, instances, and more. 

Logarithmic Function Definition

The logarithmic function is the inverse of exponentiation in mathematics. The logarithmic function definition is as follows:

For x > 0 , a > 0, and a (subscript) ≠ 1,

y= logax if and only if x = a^y

Then, the function is:

f(x) = logax

The logarithm’s base is equal to a. This is written as log base a of x. Base 10 and base e are the most commonly used bases in logarithmic functions.

Types of Logarithmic Functions

The power with which a fixed number’s base is raised to obtain a given number is represented by the logarithm. The general representation of the logarithmic function is

logax = f(x)

Typically, there are two types of logarithmic functions:

• A popular logarithmic function’s base is 10. The log 10 or log function is used to represent this function
• The natural logarithmic function’s base is e. The letters ln or loge stand for this function

• Common Logarithmic Function

The common logarithmic function, indicated as log10 or simply log, is a logarithmic function with a base of 10.

f(x) = log10x

• Natural Logarithmic Function

The natural logarithmic function is denoted by loge and is a logarithmic function to the base e (subscript).

f(x) = logex

How to Use a Logarithm Table?

To compute the logarithm of an integer, use a logarithm table. Let’s take a look at how to find a logarithm in detail.

Step 1

Recognize the notion of a logarithm. Each log table can only be used with one type of basis. The log base 10 table is the most frequent sort of logarithm table.

Step 2 

Write the number in scientific notation. For example, the number 31.62 is represented as 3.162 X 101. As a result, the 10th power has been reduced to 1. As a result, the resultant logarithmic value’s distinguishing property is 1.

Step 3 

Find the cell where the first two digits of a number are labelled on the associated row and the third digit of a number is labelled on the corresponding column header. As a result, disregard the decimal point and inspect the cell in row 31 and column 6 for the value 31.62. 0.4997 is the answer.

Step 4 

Find the value of the cell at MEAN DIFFERENCE column number 2 because it is the 4th digit of the 31.62.

To get the logarithm of the number, put the characteristic and mantissa components together.

As a result, log1031.62 = 1 + 0.5000 = 1.5000

The Mantissa is the fractional part of the number’s logarithmic form, while the Characteristic is the integer portion of the number’s logarithmic form.

Properties of Logarithmic Tables

This section contains all of the important logarithmic function properties. Let’s discuss them in detail!

Theorem 1: The total of the logarithms of two numbers, say a and b, equals the logarithm of their product. The base of both numbers should be the same.

logb(xy) = logbx + logby

Theorem 2: The division of two numbers equals the antilog of the difference of their logarithms.

In other words, the difference of the logarithms of two integers, say a and b, equals the logarithm of their division. The base of both numbers should be the same.

logb(x/y) = logbx – logby

Theorem 3: Calculating the logarithm of a number in each base can be done by calculating the logarithm of the same number in any base.

logb(xn) = n logbx

Theorem 4: The logarithm of the number raised to a power equals the index of the power multiplied by the logarithm of the number. 

logbx = logax / logab

The four logarithmic features are as follows. You’ll be able to use the logarithmic power rule, product rule, or quotient rule to rewrite a logarithmic equation.

Things to Remember

• The logarithm table is a mathematical tool for determining the value of a logarithmic function
• An exponential function can be represented in logarithmic form. In the same way, all logarithmic functions can be stated in exponential form
• You can calculate the logarithm of a number in any base by calculating the logarithm of the same number in any base
• The logarithm of the number raised to a power equals the index of the power multiplied by the logarithm of the number

Conclusion 

You would have understood logarithmic functions definition, tables, examples, and more in this article. 

There’s no disputing that logarithms are important and heavily weighted in the IIT-JEE Mains exam. You are more likely to excel in the examination with good scores if you have a thorough comprehension of the issue, including its principles and terminologies.