Lec 52 - Logarithmic Functions
Summary
TLDRThis 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
📚 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.
🔍 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.
📉 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.
📈 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.
📊 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.
Mindmap
Keywords
💡Exponential Function
💡Inverse Function
💡Logarithmic Function
💡Domain
💡Range
💡One-to-One Correspondence
💡7 Rule
💡Graph
💡Asymptote
💡Nodal Point
💡Reflection
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.
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