Lec 52 - Logarithmic Functions

Summary

This educational video delves into the inverse of exponential functions, focusing on their properties, graphs, and the domain and range of these inverse functions. It introduces the concept of logarithmic functions as the inverse of exponential functions, explains the '7 rule' for understanding their one-to-one correspondence, and illustrates how to determine the domain and range of basic and derived logarithmic functions. The video also demonstrates the reversibility of logarithmic and exponential functions through examples and discusses the graphical representation of these inverse functions, emphasizing the reflection along the line y=x.

Takeaways

• 📚 The video discusses the inverse of exponential functions, focusing on their properties, graphs, and the domain and range of these inverse functions.
• 🔍 An exponential function is defined as f(x) = a^x, with conditions that 'a' must be greater than 0 and not equal to 1 to ensure the function is one-to-one and has an inverse.
• 🔄 The inverse of an exponential function is a logarithmic function, denoted as log_a(x), and is defined as the inverse of a^x.
• 📉 The domain of the exponential function (a^x) is all real numbers, and its range is from 0 to infinity, not including negative values.
• 📈 The range of the logarithmic function is all real numbers, while its domain is from 0 to infinity, not including 0 and negative values.
• 🔢 The '7 rule' is introduced as a memory technique to recall the one-to-one correspondence between logarithmic and exponential functions: if y = log_a(x), then x = a^y.
• 📝 The script verifies that the logarithmic function is indeed the inverse of the exponential function by applying the rules f(f^-1(x)) = x and f^-1(f(x)) = x.
• 🤔 The importance of understanding the domain and range of both exponential and logarithmic functions is emphasized for problem-solving and function composition.
• 📉 The video explains how to determine the domain of more complex functions composed with logarithms, such as log_a(1-x), by ensuring the argument of the logarithm is within its valid domain.
• 📈 The script demonstrates the reversibility of logarithmic and exponential functions with examples, showing how to find values and solve equations using their inverse relationship.
• 📊 The video concludes with a discussion on graphing the inverse functions, describing the process of reflecting the graph of the exponential function across the line y=x to obtain the logarithmic function's graph.

Q & A

What is the main focus of the video?

The video focuses on the inverse of exponential functions, discussing their properties, graphs, and the relationship between domain and range of these inverse functions.

What is an exponential function?

An exponential function is a function of the form f(x) = a^x, where 'a' is a base that is greater than 0 and not equal to 1.

Why is the base 'a' in an exponential function restricted to be greater than 0 and not equal to 1?

The restriction on 'a' being greater than 0 and not equal to 1 ensures that the function is one-to-one and has an inverse. If 'a' were 1, the function would be constant and not interesting to study as it would not have an inverse.

What is the definition of a logarithmic function?

A logarithmic function to the base 'a' is defined as y = log_a(x), where 'x' is the argument of the function and it is the inverse of the exponential function f(x) = a^x.

What is the '7 rule' mentioned in the video?

The '7 rule' is a mnemonic to remember the one-to-one correspondence between logarithmic and exponential functions. It states that if y = log_a(x), then x = a^y.

How can you verify if a function is the inverse of another function?

You can verify if a function is the inverse by checking if f(f^-1(x)) = x and f^-1(f(x)) = x, where f^-1 represents the inverse function.

What are the domain and range of the logarithmic function?

The domain of the logarithmic function is (0, ∞), meaning it cannot have negative values or zero. The range of the logarithmic function is the entire set of real numbers.

How does the domain of a function affect its graph?

The domain of a function determines the set of all possible input values for which the function is defined. This directly affects the graph of the function, as the graph will only include points that correspond to values within the domain.

What is the relationship between the graph of an exponential function and its inverse logarithmic function?

The graph of the inverse logarithmic function is the reflection of the original exponential function along the line y = x. This means the graphs are mirror images of each other across this line.

How can you use logarithmic functions to solve problems involving exponents?

You can use the property that a^(log_a(x)) = x to solve problems. For example, if you know a^y = x and you want to find y, you can use log_a(x) to find that y = log_a(x).

What is the significance of the base 'a' in the logarithmic function in relation to the domain?

The base 'a' of the logarithmic function determines the range of the original exponential function, which in turn becomes the domain of the logarithmic function. This means that the logarithmic function is defined for arguments greater than zero.

Outlines

00:00

📚 Introduction to Inverse Exponential Functions

This paragraph introduces the topic of the video, which is the inverse of exponential functions. It explains that if a function is one-to-one, finding its inverse is straightforward. The video will focus on the properties, graphing, and domain and range of the inverse functions of exponential functions. The definition of an exponential function is also provided, with conditions on the base 'a', such as 'a' must be greater than 0 and not equal to 1. The paragraph concludes by stating that the inverse of an exponential function is a logarithmic function, defined as y = log_a(x), and emphasizes the importance of understanding the domain and range of these functions.

05:02

🔍 Verifying the Inverse Relationship Between Exponential and Logarithmic Functions

The second paragraph delves into verifying the inverse relationship between exponential and logarithmic functions. It discusses the need to ensure that the logarithmic function is indeed the inverse of the exponential function by applying two rules: f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. The paragraph explains the domain and range of both functions, highlighting that the domain of the exponential function is the entire real line, while the range is from 0 to infinity. Conversely, the domain of the logarithmic function is from 0 to infinity, and its range is the entire real line. The summary also introduces the '7 rule' as a mnemonic for remembering the one-to-one correspondence between the exponential and logarithmic functions.

10:11

📉 Determining the Domain of Logarithmic Functions

This paragraph focuses on determining the domain of logarithmic functions, using examples to illustrate the process. It explains that the domain of a logarithmic function is restricted to values greater than 0, as the argument of the logarithm cannot be negative or zero. The paragraph provides a step-by-step analysis for finding the domain of a specific function, f(x) = log_4(1-x), and another function, g(x), which involves a more complex argument. It emphasizes the importance of ensuring that the function's domain is valid before attempting to solve problems involving logarithms.

15:14

📈 Demonstrating the Reversibility of Logarithmic Functions

The fourth paragraph demonstrates the reversibility of logarithmic functions through examples. It explains how to use the logarithmic function to reverse an exponential expression, showing that 3^y can be rewritten as 3^(log_3(x)) to simplify calculations. The paragraph also discusses how to solve for an unknown variable in an exponential equation by taking the logarithm of both sides. It concludes with an example of finding the base-3 logarithm of 1/9 by recognizing it as 3^(-2), thus simplifying the problem.

20:14

📊 Graphing the Inverse Exponential (Logarithmic) Functions

The final paragraph discusses the graphical representation of the inverse exponential, or logarithmic, functions. It begins by reviewing the graphs of exponential functions and then describes how to reflect these graphs across the line y=x to obtain the graphs of their inverse functions. The paragraph provides a visual explanation of how to translate points from the original function to its inverse and illustrates the process for both cases where the base 'a' is between 0 and 1, and where 'a' is greater than 1. It concludes by summarizing the video's content and预告ing the next video, which will explore the application of logarithmic functions in solving mathematical problems.

Keywords

💡Exponential Function

An exponential function is a mathematical function of the form f(x) = a^x, where 'a' is a positive real number not equal to 1. It is central to the video's theme as it is the original function whose inverse is being discussed. The video script uses the exponential function to introduce the concept of its inverse, the logarithmic function.

💡Inverse Function

An inverse function reverses the effect of the original function. In the context of the video, the inverse of an exponential function is a logarithmic function. The script explains that if a function is one-to-one, it has an inverse, which is a crucial concept for understanding the properties of exponential and logarithmic functions.

💡Logarithmic Function

A logarithmic function is the inverse of an exponential function and is defined as y = log_a(x), where 'a' is the base and 'x' is the argument. The video script defines the logarithmic function and discusses its properties, emphasizing its relationship with the exponential function as its inverse.

💡Domain

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. The video script explains the domain of logarithmic functions, noting that they cannot include negative values or zero, and relates this to the range of the exponential function.

💡Range

The range of a function is the set of all possible output values (y-values) the function can produce. The video script discusses the range of both exponential and logarithmic functions, illustrating how the range of the exponential function is the domain of its inverse, the logarithmic function, and vice versa.

💡One-to-One Correspondence

A one-to-one correspondence is a relationship between two sets such that each element of one set is uniquely paired with an element of the other set. The script uses this concept to explain the relationship between exponential and logarithmic functions, stating that the logarithm of a number is the exponent to which the base must be raised to produce that number.

💡7 Rule

The '7 Rule' is a mnemonic used in the script to remember the relationship between logarithmic and exponential functions. It suggests that if you start with the base 'a', apply the function (exponential or logarithmic), and then reverse the process, you end up with the original number. The script uses this rule to help memorize the inverse relationship between the two functions.

💡Graph

The graph of a function is a visual representation of the relationship between the input and output values. The video script describes how to graph the inverse function (logarithmic) by reflecting the graph of the original function (exponential) across the line y=x, demonstrating the mirror image relationship between the two.

💡Asymptote

An asymptote is a line that a curve on a graph approaches but never intersects. The script mentions that the logarithmic function does not have an asymptote on the y-axis, unlike some other functions, and this is an important characteristic when graphing logarithmic functions.

💡Nodal Point

A nodal point is a point on a graph where two branches of a curve intersect. The video script refers to the nodal point when discussing the graph of the logarithmic function, indicating that the graph passes through this point and is an important feature in understanding the shape of the graph.

💡Reflection

In the context of graphing inverse functions, reflection refers to the process of flipping the graph of a function across the line y=x to obtain the graph of its inverse. The script explains that the graph of the logarithmic function is the reflection of the exponential function's graph, which is key to understanding the visual representation of inverse functions.

Highlights

Introduction to the inverse of exponential functions and their properties.

Explanation of when a function is one-to-one and thus has an inverse.

The definition of an exponential function with base 'a' and its conditions.

The concept of a logarithmic function as the inverse of an exponential function.

The standard form of a logarithmic function and its relation to the exponential function.

Understanding the domain and range of the logarithmic function.

The '7 rule' for remembering the one-to-one correspondence between logarithmic and exponential functions.

Verification of the logarithmic function as the inverse of the exponential function using the '7 rule'.

The importance of understanding the domain and range for both exponential and logarithmic functions.

How to determine the domain of a function derived from a logarithmic function.

Example of finding the domain of a function involving a logarithm with base 4.

Explanation of the reversibility of logarithmic and exponential functions with examples.

Demonstration of how to use logarithmic functions to solve for unknowns in exponential equations.

The graphical representation of the inverse function and its relation to the original exponential function.

How to draw the graph of the logarithmic function as a mirror image of the exponential function.

Differentiating the graphs of inverse functions based on the value of 'a' in the exponential function.

The practical application of logarithmic functions in simplifying mathematical problems.

Conclusion and preview of the next video focusing on the formulation of mathematical problems using logarithmic functions.

Transcripts

So, in this video we are going to look  at the inverse of exponential function.  

In the last video we have seen the inverse  of a general function and we have concluded  

that if the function is one-to-one, then the  finding the inverse of a function is very easy.  

So, let us focus on inverse of  exponential function in this video  

and see its properties graph or how it  is graphed and a various other properties  

about domain and range of these inverse  functions for exponential functions.  

So, let us recall our notion of exponential  function, we started with a function which is  

a function will be called as exponential function  if it is written in the form f of x is equal to  

a raise to x where there were some conditions  on a, for example, a should be greater than 0  

and a cannot be equal to 1, a greater than 0 is a  typical condition which we need because otherwise  

we have to deal with complex random, complex  variables which is out of scope of this course.  

So, we are putting a to b greater than 0 and a not  equal to 1 is the condition because if you put a  

is equal to 1, then f of x is equal to 1 raise  to x which is 1 for all of them, so it is not  

an interesting function to study. So, whenever  these conditions are enforced we know that our  

exponential function f of x equal to a raise  to x is one-to-one and because every one-to-one  

function has the inverse this function also has  the inverse, there is nothing special about it.  

And that inverse we will define as logarithmic  function. So, naturally since we are talking  

about exponential function with base a so we will  talk about logarithmic function with base a.  

So, here is a definition  of a logarithmic function.  

The definition says that the logarithmic  function to the base a in the standard form is  

given by y is equal to log to the base a of x.  So, remember this function is represented by  

log to the base a and x is the argument of the  function, so this is the definition of a function  

or this is replacing f, f inverse and then x is  the argument and we are plotting it (acro) along y  

axis and is defined to be the inverse of the  function f of x which is equal to a raise to x.  

So, f inverse of x is actually log to the base  a of x, is this simple. So, now we need to  

understand what will be the domain and codomain  or range of this function that is an important  

thing that we need to understand. So, in order  to that let us try to devise some rule so that  

we will have a track of what is exactly happening  when we are talking about logarithmic function  

and how it is related to exponential function. So, there is a one to one correspondence between  

logarithmic function and an exponential  function which is expressed by this relation  

y is equal to log to the base a of x if  and only if x is equal to a raise to y  

or for more precision you can write this as  log to the base a of x is equal to y then  

you can actually virtually assume this  7 rule that is you start from the base,  

go to the right hand side and come back that  means what we are saying is you start with a,  

go to the right hand side, that right hand side  is raise to the power and that should give you  

x, that is what this rule is. So, this is simple technique to remember  

known as 7 rule. So, you can use this 7 rule to  memorize the one-to-one correspondence between log  

and the exponential function. You can  easily see that by definition if I write x  

is equal to a raise to y, then I want to  know the value of y, I should be able to  

get it by taking the log of this function x. So, this is the mathematical definition of our  

logarithmic function. To make this mathematical  definition precise we need to understand some  

prototypes that is whether this function we  have defined it to be the inverse of f but  

whether this function is actually the inverse of  f or not that is what we need to figure out.  

So, as stated earlier we can actually check  these two rules f inverse f of f inverse of  

x should be x and f inverse of f x should also  be equal to x. So, what is f of f inverse of x?  

As I mentioned earlier f inverse of x is nothing  but log of x to the base a and f is a raise to x  

so you just substitute a raise to f inverse of  x. What is that? a raise to log a of x. Now,  

what this should be? You use this one to one  correspondence from here to here and here to  

here and you will get this to be equal to x. In a similar manner you can apply it to f of x  

and f inverse, so f inverse of f of x is log to  the base a of f of x but what is f of x? It is  

a raise to x and therefore log to the base  a of a raise to x should be equal to x.  

Now, in order to understand this completely I  need to understand the domain of log function  

and range of log function and the range of log,  range of exponential function and the domain of  

exponential function. So, let us understand this  particular thing. We have already seen what is the  

domain of a raise to x, so we already know domain  of a raise to x because x can be entire real line  

and then it maps this domain onto the range of a  raise to x that range cannot take negative values,  

this is what we have seen when we studied. So, it was 0 to infinity, so this should be clear  

before going to the range of log function. So, if  at all the logarithmic function is to be defined,  

this if you recollect this should  become domain of log to the base a  

and this should become the range of log  to the base a, so the this is the crux of  

the definition of inverse. So, when  this is satisfied you are done.  

So, essentially your log function will be  defined from 0 to infinity to real line.  

That means in the domain it cannot have negative  values, it cannot have 0 as well and in the range  

it will have the entire real line that is what  is written here in this case that is domain of  

log to the base a is actually range of a raise to  x which is 0, infinity and domain of a raise to x  

is actually the range of log to the base a  which is real line, the entire real line.  

These are the two important points which will help  you in understanding the domains of the functions  

which are derived from these functions that is  logarithmic functions or exponential functions.  

So, these, all these things you should always  remember the valid ranges and domains of the  

function. So, this completes our verification  that logarithm function the way we have defined  

is actually an inverse of exponential function.  Once the verification is complete let us dwell  

more and find the domain of the derived  functions, derived, by derived functions means  

composition of basic logarithmic function. For example, let us take an example of f x which  

is log to the base 4 of 1 minus x. Now, log to the  base 4 is actually a function which has a domain.  

What is the domain of this function? The domain  of this function is actually 0 to infinity.  

Now, that means the argument that is supplied  to this function log to the base 4 cannot be 0,  

or it cannot be a negative value. So, based on  this understanding from the definition of our  

log function you can look at this function  which is f of x and look at the argument  

of the function 1 minus x. According to this definition  

1 minus x must be strictly greater than 0. This  will happen if and only if my 1 is greater than x,  

1 is greater than x and because 1  minus x needs to be greater than 0  

can x be less than 0, if you  look at x to be less than 0, 1  

minus x will actually be greater than 0. So,  the only condition that we require over here  

is my function should be defined that is  domain of this function f should be equal to,  

it cannot include 1, 1 to, it is not 1 to  infinity, this is how we commit mistakes.  

So, domain of f is x should always be less than 1  that means the domain of this function should be  

here minus infinity to 1 and it cannot go  beyond 1 this is what our understanding is  

about this function. Now, let us  go and enhance our understanding  

in finding the domain of a function which is  slightly more complicated than this function.  

So, our question is to find the domain of this  function g. In order to find the domain of this  

function g, let us first understand what is  the domain of the function log to the base 3.  

Now, this function is defined when the  argument given is between 0 to infinity.  

So, now I want the argument  of this function which is this  

gx to be between 0 to infinity. So, what I  should do is I want this 1 plus x upon 1 minus x  

trapped between 0 to infinity that means it should  be greater than 0. Now, when this can happen?  

So, naturally let us split the real line into some  parts x is not equal to 1 is already given to you,  

so x cannot take the value 1, this is a point 0,  this is a point 1, let, for safety let us put the  

point minus 1 as well here. And now x cannot be  equal to 1, so this point is actually deleted,  

so this point cannot be there. Then, 1 minus x  should, if 1 minus x is greater than 0 that means  

my x is less than 1 the function is defined. So, I have this in the similar manner minus  

infinity to 1 but let us not go for minus infinity  because there is in the numerator there is 1 plus  

x, so this 1 plus x, it can become, it can take  a negative value when x is less than minus 1  

and if x is less than minus 1 this 1 minus  x will become positive. So, I have to rule  

out that part as well. So, this minus 1 to 1 is  rule, minus infinity to minus 1 is ruled out,  

minus 1 will give me the value 0 so  minus 1 is also ruled out and therefore  

I am only left with the interval of  this form which is minus 1 to 1.  

So, based on the arguments and based on this  domain I know that the domain of this function is  

valid only between minus 1 to 1. Now, you may say  why not 0? 0 will not cause any problem because  

if you look at the function, if you substitute x  is equal to 0 you will get log to the base 3 of 1  

which is a positive number and therefore it is  well defined. So, the domain of this function is  

nothing but minus 1 to 1, this is how we need  before trying to solve any problem related to  

logarithms we need to first verify whether it  is, what the problem that we are willing to  

solve is defined in a proper domain or not. Most of the times when you try to formulate a  

problem the problem may not be defined in a  proper domain and then solving that problem  

is a meaningless exercise. So, just to ensure that  always your problem is defined in a valid domain.  

So, this ends the verification of this. Now,  let us take one more example which will actually  

help you in understanding the reversibility  of log and exponential function.  

So, here is an example where we are actually  demonstrating the reversibility of a log function  

or the inverse of a log function. So,  y is equal to log to the base 3 of x.  

We assume that everything is well defined  and this x belongs to 0 to infinity.  

In that case this y will belong to the real line  and if I want to write 3 raise to y then I will  

write 3 raise to y as 3 raise to log 3 of x. By definition, by definition this function is the  

inverse of the log function. Therefore,  you will get this to be equal to x  

and therefore your 3 raise to y is equal to  x. Now, how this helps in your calculations?  

Suppose, you know some number 1.3 raise to 2  is equal to m and you want to identify this m.  

Then you can actually take the log of this  function, log of this function which is the  

inverse of this and which will be  equal to log to the base 1.3 of m  

and if you equate these two what you get here is  2 being equal to log to the base 1.3 of m.  

Why is it so? Because 1.3 square we have  taken the log so this is like a raise to  

x and you are simplifying it. So, a raise to  x a raise to, a raise to log to the base a  

of x is actually x. So, you will get the number 2  naturally. So, this is how the log thing helps.  

And here what the, the fact that we have  used is a raise to u is equal to a raise to v  

for a greater than 0 and a not  equal to 1 implies u is equal to v.  

If you use this fact and you are asked  to find the log to the base 3 of 1 by 9,  

then you can easily find. Let us see how. So,  you start with log to the base 3 of 1 by 9.  

Now, you look at this 9 and 3. If you  look at 3 square that will give you 9  

isn't it and that also implies 3  raise to minus 2 will give me 1 by 9.  

So, I will simply use the fact that log to the  base 3 of 3 raise to minus 2 is 1 by 9.  

So, but this is an inverse function, this  is like 3 raise this particular thing  

is like 3 raise to x, log of 3 raise to x is  again going to be x, so you will get minus 2  

to be the answer, there this is how you  can solve some problems very easily when  

you can identify the base is actually  multiple of this particular argument.  

So, this is the use of log we will deal with it  in more detail when we will solve the problems on  

logarithms. Now, for a moment we have identified  what is the inverse function of our exponential  

function, it is logarithmic function to  the same base as exponential function.  

Let us try to look at the graph of the inverse  function that is graph of f x equal to log to  

the base a of x. How will it look like? If you  remember the graphs of exponential functions,  

the graphs of exponential functions were  having two discriminations, like if you take  

a the line from 0 to infinity, then there was  some split at 1 and from 0 to 1 when there is,  

the value of a lies in 0 to 1, the graph was  different and from this side onwards that is a  

is greater than 1 the graph was different. So, let us first imagine those graphs and let  

us recollect from the previous video what was  the interpretation of the graph of the inverse  

function. If you recollect from the previous  video, the graph of the inverse function  

is nothing but the reflection of the original  function f along the line y is equal to x  

or the mirror image of the function. So, let us look at the exponential  

function first when 0 is less than a  is less than 1 and a is greater than 1.  

So, this is the graph when 0 is less than a  is less than 1. Now, I have made it big enough  

so that you can understand better and the  blue line is the line y is equal to x.  

Now, if I want to translate the mirror image  of this function how will I translate?  

Let us take one point so let  us take a point 0, 1 over here,  

the translation of that point will be 1, 0  over here and then take this point over here,  

I should not draw any point here because it may  confuse you so the translation of that point in  

this zone is a point over here and a point over  here, and similarly you go on translating and  

connect the two lines. For example, here if I go on  

translating this point then the translation  will actually go to some place over here  

and if you take one more point over here then the  translation will actually go to the other quadrant  

which is 2 units below this and over  here. So, the graph of this function will  

actually look something like this, it  will pass through the same nodal point  

and it will pass through this and then on y axis  it will be very flat, very close to the y axis and  

so on, so this is how the graph of the function  will look like because it is a mirror image.  

So, this is how it will look like, it is not an  asymptote but because the graph paper is over I am  

not able to draw. In a similar  manner this is the case when 0  

less than a less than 1. So, I have  drawn the graph in the next sheet  

which is a green line you can see this green  line actually matches with this green line,  

I have slightly shifted the graph paper  in order to have a better visibility.  

Now, you can actually see this is the original  function, this is the new inverse function and  

this is the line y is equal to x, so you can  see the correspondence of the inverse function  

with respect to the original function, all this is  possible because our function is one-to-one.  

Now, if you look at, again look at the graph  of a function where a is greater than 1  

then this is the graph of a function here  there are no overlaps, so it is relatively  

easy to draw the graph. For example, I can choose  this point over here if I go one unit from here  

I should get something like this here so it  is a reflection along x axis, so it will be  

relatively easy to draw the graph here, this point  reflected here that point will be reflected here  

and then I can draw that, I can join the curve  like this and it will be exact mirror image of the  

original function and it will be going  close to this particular function.  

So, roughly this will be the graph, I have drawn  the full proof graph on the next graph paper which  

is here. So, now you can easily visualize the  graphs of both the functions, let us zoom out and  

see all of them together all 4 graphs together.  So, these are all 4 graphs handled together,  

so my graph actually looks like this graph  for both the cases, so this is how it is easy  

to draw the graphs of inverse functions once  we know the graph of the original function.  

In the next, this is, that is all for this video.  In the next video what we will see is we will try  

to use our knowledge of logarithmic functions and  try to see how the formulation of a mathematical  

problems becomes easy when we consider logarithmic  functions, even though there is a limitation that  

logarithmic function is defined only from 0 to  infinity not on the real line. Thank you.