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Logarithmic Functions

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Last updated date: 19th May 2024
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Introduction

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

Log(M/N) = logb  (M) – log(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 – log9 = log6 (54/9) = log6 6 = 1

The Power Rule

Log(Mp ) = p log(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

Log1 = 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) = logx + logy

logb (x/y) = logb x – logy

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 = log25

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 a= 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.   

FAQs on Logarithmic Functions

1. What is the parent function of logarithmic functions?

Logarithms are used to explore properties of exponential functions and to solve exponential functions. For any log parent function is written as f(x) = logb x. for example, g(x) = log4  x corresponds to another family of functions then h(x) = log8 x. the example graphs the common log: f(x) = log x. you can change the log into exponential. Finding the parent of logarithmic functions is valuable in calculus where it is used to calculate the slope of certain functions and the area bounded by certain curves.

2. What does a logarithmic function look like?

While speaking a logarithmic function is said as “the log, base a, of x.” the logarithmic function is the inverse of the exponential function, so logarithms can only be expressed by using its exponential form. It is the same operation by thinking “a to the power y equals to x”. The common logarithmic function is written as y = log x and natural logarithmic function is written as f (x) = log e x. Here ‘e’ is any number whose value is greater than zero, but excludes ‘1’. An interesting fact is that the logarithmic function is always on the positive side of the y-axis. It never crosses over to reach the negative side of the y-axis.

3. Explain the properties of Logarithmic Functions.

Properties of Logarithmic Functions are mentioned below. 

  • Product Rule

log AB = log A + log B

Multiplying two numbers having the same base and adding the exponents. 

  • Quotient Rule 

log A/B = log A – log B 

Dividing two numbers with the same base and subtracting the exponents

  • Zero Exponent Rule 

log 1 = 0.

  • Power Rule 

Log Ap = P log A

Multiply the exponents and raise the exponential expression to a power. 

  • Change of Base Rule 

logb (x) = log₁₀ x / log₁₀ b

4. From where can I learn Logarithmic Functions?

You can learn Logarithmic Functions from Vedantu’s free online learning platform. With detailed explanations and solved examples, we provide you with the best study materials to study Logarithmic Functions. You can use our website or download the mobile app to get access to the notes on Logarithmic Functions for absolutely free. You don’t even have to pay any registration fees on Vedantu. Our study resources are accessible to all the students around the globe. Apart from Logarithmic Functions, you can learn other concepts of math as well only on Vedantu. We have notes on concepts like probability, exponents, trigonometry, integration, and a lot more.

5. Is it important to study Logarithmic Functions?  

Yes, it is necessary to learn Logarithmic Functions as they are an important concept of mathematics. These functions are related to the exponential functions, which is also a crucial concept. So, by learning logarithmic functions, you can solve logarithms and exponential equations. These functions also help in solving calculus problems and play a vital role in practical applications of economics. So, learning logarithmic functions will improve your understanding of various other concepts of math and economics as well.