Logarithmic Functions

In mathematics, scholars used logarithms to change division and multiplication problems into subtraction and addition problems, even before the discovery of calculus. In logarithm to get a certain number, the power is raised to some number which is typically a base number. Logarithmic functions are inverse of exponential functions and you can express any exponential function in logarithmic form. Likewise, all logarithmic functions are rewritten in exponential form. When you want to work with very large numbers, logarithms are useful in manipulating numbers of a much more manageable size. In this section, we will further discuss the definition, formula, and functions in detail along with some examples.

Logarithmic Functions Definition

A logarithm is an exponent that is written in a special way. For example, we know the following exponential equation is true 3² = 9. Here the exponent is 2 and the base is 3. In the logarithmic form, we will write the equation as log3 9 = 2. In words we say this as “the logarithm of 9 to the base 3 is 2”. Here we have effectively moved the exponent down to the mainline. This was done to make divisions and multiplications easier but still, logarithms are very handy in mathematics.

A logarithmic function is defined as f (x) = logb x. Here the base of logarithm is b. base and base 10 are the common bases we use which we meet in logs to base 10 and natural logs.In real life, there are many logarithmic applications like in electronics, earthquake analysis, acoustic, and population prediction. 

Introduction to Common and Natural Logarithmic Function

Common Logarithmic Function

A logarithm with base 10 is a common logarithm. In our number system, there are ten bases and ten digits from 0-9, here the place value is determined by groups of ten. You can remember common logarithms with the one whose base is common as 10. 

Natural Logarithmic Function

A natural logarithm is different. When the base of the common logarithm is 10 the base of a natural logarithm is number e. Although e represents a variable it is a fixed irrational number that equals 2.718281828459. Sometimes e is also known as Euler’s number or Napier’s constant. The letter e is chosen to honor mathematician Leonhard Euler. e looks complicated but is rather an interesting number. The function f (x) = loge x has multiple applications in business, economics, and biology. Therefore, e is an important number.

Properties of Logarithmic Functions

The Product Rule

Logb (MN) = logb (M) + logb (N)

This property denotes that the logarithm of a product is the sum of the logs of its factors.

Multiply two numbers having same base, then add the exponents

Example: log 20 + log 2 = log 40

The Quotient Rule

Logb  (M/N) = logb  (M) – logb  (N)

This property denotes that the log of a quotient is the difference of the log of the dividend and the divisor.

Divide two numbers having the same base and subtract the exponent.

Example: log6 54 – log6  9 = log6 (54/9) = log6 6 = 1

The Power Rule

Logb  (Mp ) = p logb  (M)

The property denotes that the log of a power is the exponent times the logarithm of the base of the power.

Raise an exponential expression to power and multiply the exponents

4 log5 (2) 

= log5 (24 ) 

= log5 (16)

The Zero Exponent Rule

Loga  1 = 0

Change of Base Rule

logb (x) = x / b or logb (x) = log10 x / log10 b

some other properties of logarithms are as follows:

logb (xy) = logb  x + logb  y

logb (x/y) = logb x – logb  y

logb (xr ) = rlogb x

if logb x = logb y therefore x = y

Examples of Logarithmic Functions

1. Write the exponential equation in logarithmic form

1. 52 = 25

2. 4-3  = 1/64

3. (1/2)-4   = 16

Solution: a. 52  = 25  becomes 2 = log5  25

b. 4-3  = 1/64 becomes -3 = log4  (1/64)

c. (1/2)-4  = 16 becomes -4 = log(1/2)  16

2. Write logarithmic equation in exponential form

a. log6 36 = 2

b. loga m = p

Solution: a. log6 36 = 2 become 62 = 36

    b. loga m = p becomes ap  = m

3. Solve the following equations

a. log7 49 = y

b. log2  (1/8) = y

Solution: a. log7 49 = y becomes 7y  = 49

          since, 49 = 72            7y  = 72 

          so, y = 2

      b. log2 (1/8) = y becomes 2y = 1/8

          since, 1/8 = 2-3     2y = 2-3 

          so, y = -3

How to Learn Logarithmic Functions?

When you start learning logarithmic functions, they might be a little difficult to understand. However, with the right study materials and skills, you can build a strong grasp of logarithmic functions in no time. Below are some tips and tricks to help you learn these functions with ease. 

• Before you start learning logarithmic functions, you have to first understand the difference between logarithms and exponents as they are quite similar. 

• When you are studying logarithmic functions, you will come across terms like a natural log and a common log. You have to go through these terms thoroughly to get a better understanding of logarithmic functions. 

• Logarithmic functions have some properties that you have to learn before solving equations. You have to understand these properties and learn how to apply them to different questions. 

• Use solved examples provided in your textbooks to understand how to solve questions based on logarithmic functions. You can also refer to Vedantu to find solved questions on logarithmic equations. 

• Practice as many questions as you can to practice the concept of logarithmic functions and become more proficient in solving these questions. 

• Learn other concepts in which logarithms are used to understand the applications of logarithmic functions in different areas. 

• To understand the concept of logarithmic functions, you can use Vedantu’s e-learning platform that provides you with the definitions, properties, and applications of these functions for absolutely free. For more convenience, you can use our services on your smartphone by downloading the Vedantu app from the play store. 

• Go through different sample papers and previous year question papers to get an idea of the exam pattern, difficulty level and types of questions based on logarithmic functions that can be asked in your math exam. 

• Get access to all the study materials such as revision notes, textbook solutions, reference books, etc. to learn more about logarithmic functions and enhance your knowledge of the concept. 

• Once you have read the properties and definitions of logarithmic functions, you can start solving practical questions to check your knowledge and see whether you have understood logarithms or not. 
  
  

Importance of Learning Logarithmic Functions 

Learning logarithmic functions is really important for all mathematics students. These functions are an essential part of the math syllabus. Logarithmic functions can be applied in many fields for different purposes. That is why learning this concept has become a necessity for students. Below are some other reasons why you should learn logarithmic functions: 

• Logarithmic functions make it easier for you to understand calculus, exponents, and various other complicated concepts of mathematics. 

• Logarithmic functions allow you to solve problems based on exponents, integrals, differentiation, etc. and improve your understanding of these topics. 

• Once you have learned the properties of logarithmic functions, you will be able to apply them in different questions. 

• If you know the applications of logarithmic functions perfectly, you can score significantly well in your math tests. 

• Learning logarithms will help you differentiate between logarithmic functions and exponential functions.