NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.4 Continuity and Differentiability - PDF Download

NCERT solutions for class 12 maths chapter 5 exercise 5.4 Continuity and Differentiability addresses the essential concepts of exponential and logarithmic functions. To solve complicated problems of differentiability and get an in-depth knowledge of concepts, you must follow a precise understanding of exponential and logarithmic functions. In order to get a clear understanding of questions included in ex 5.4, you can download the NCERT solutions for class 12 maths chapter 5 ex 5.4 developed by an academic team of mathematics at eSaral.

Class 12 maths chapter 5 exercise 5.4 NCERT solutions consist of a total of 10 questions related to plotting a graph for the exponential and logarithmic functions. Several examples are also covered in ex 5.4 class 12 maths NCERT solutions that deliver the fundamentals of this topic. It is effective to develop a thorough understanding of the fundamentals of concepts by solving sums that are included in this exercise.

Ex 5.4 class 12 maths chapter 5 is provided in PDF version for students to promote error-free learning. You can download these PDFs for free from the official website of eSaral and practice questions to prepare for final exams. 

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Topics Covered in Exercise 5.4 Class 12 Mathematics Questions

Ex 5.4 class 12  maths ch 5 covers exponential and logarithmic functions of differential. Let’s get a detailed explanation of this topic to solve exercise questions.

1. Exponential and Logarithmic Functions

Exponential Functions

A mathematical function with the form y = f(x) = bx, where "x" is a variable and "b" is a constant that is referred to as the function's base and such that b > 1, is called an exponential function. The transcendental number e is the exponential function base that is most frequently used.

The exponential function can be represented as y = ex by using "e" as the basis. We refer to this as the natural exponential function. Conversely, the exponential function with base 10 is the common exponential function.

Logarithmic Functions

If the inverse of the exponential function is known, the logarithmic function can be represented as follows:

Let b > 1 be a real number such that the logarithm of a to base b is x if b^x = a.

Logarithm of a to base b is denoted by log_b a. Thus log_b a = x if b^x = a.

In other words, we may recognise the logarithm as a function from positive real numbers to all real numbers by setting a base b > 1. We refer to this function as the logarithmic function and is defined by

                      $\log _b: \mathbf{R}^{+} \rightarrow \mathbf{R}$

$x \rightarrow \log _b x=y$ if $b^y=x$

 if the base b = 10, we say it is common logarithms and if b = e, then we say it is natural logarithms. Often natural logarithm is denoted by ln.

Properties of Exponential and Logarithmic Functions

The following is a list of some of the most important features of the exponential functions:

(1) Domain of the exponential function is R, the set of all real numbers.

(2) Range of the exponential function is the set of all positive real numbers

(3) The point (0, 1) is always on the graph of the exponential function (this is a restatement of the fact that b0 = 1 for any real b > 1).

(4) Exponential function is ever increasing; i.e., as we move from left to right, the graph rises above.

(5) For very large negative values of x, the exponential function is very close to 0. In other words, in the second quadrant, the graph approaches the x-axis (but never meets it).

The following is a list of a few important points to keep in mind when using the logarithm function to any base b > 1.

(1) We cannot make a meaningful definition of logarithm of non-positive numbers and hence the domain of log function is R+ .

(2) The range of the log function is the set of all real numbers.

(3) The point (1, 0) is always on the graph of the log function

(4) The log function is ever increasing, i.e., as we move from left to right the graph rises above.

(5) For x very near to zero, the value of log x can be made lesser than any given real number. In other words in the fourth quadrant the graph approaches the y-axis (but never meets it). 

2. Theorems

(1) The derivative of e^x w.r.t., x is ex ; i.e., $\frac{d}{d x}\left(e^x\right)=e^x$

(2) The derivative of log x w.r.t., x is 1/x ; i.e., $\frac{d}{d x}(\log x)=\frac{1}{x}$

Tips for Solving Exercise 5.4 Class 12 Chapter 5 Continuity and Differentiability

Our subject experts of eSaral have provided some useful tips for solving ex 5.4 class 12 maths chapter 5.

1. To develop clear understanding, students have to carefully review all definitions, theorems, and formulas while studying these functions and their properties.
2. Through interactive visuals like graphs students can acquire precise understanding of difficult and challenging topics.
3. Students must solve each question from the ex 5.4 class 12 maths NCERT solutions to score good marks in exams.
   

Importance of Solving Ex 5.4 Class 12 Maths Chapter 5 Continuity and Differentiability

There are numerous benefits of solving questions of ex 5.4 class 12 maths solutions. Some of the essential benefits are provided below.

1. All the important topics of exponential and logarithmic functions are explained by expert teachers of eSaral to provide you a deep understanding of questions included in ex 5.4 class 12 maths chapter 5. 
2. It is also beneficial to prepare for these kinds of questions in order to get in-depth fundamental knowledge for further classes and competitive exams.
3. Solving questions in NCERT solutions will help you to promote step by step learning to solve exercise 5.4 questions.
4. NCERT solutions PDF for class 12 maths chapter 5 ex 5.4 provides you answers in downloadable format that will help you to practice questions at your own pace. 
   

Frequently Asked Questions

Question 1. What is a natural exponential function?

The exponential function can be represented as y = ex by using "e" as the basis. We refer to this as the natural exponential function.

Question 2. What is a logarithmic function?

Logarithm function can be identified as a function from positive real numbers to all real numbers by setting a base b > 1. This function is called logarithmic function and is defined by $\log _b: \mathbf{R}^{+} \rightarrow \mathbf{R}$