THE CALCULUS OF LOGARITHMIC FUNCTIONS THE CALCULUS OF LOGARITHMIC FUNCTIONS.
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THE CALCULUS OF THE CALCULUS OF LOGARITHMIC FUNCTIONS LOGARITHMIC FUNCTIONS
Transcript of THE CALCULUS OF LOGARITHMIC FUNCTIONS THE CALCULUS OF LOGARITHMIC FUNCTIONS.
THE CALCULUS OF THE CALCULUS OF
LOGARITHMIC FUNCTIONSLOGARITHMIC FUNCTIONS
PREREQUISITESPREREQUISITES
This chapter is generally the start of the Calculus II This chapter is generally the start of the Calculus II curriculum.curriculum.
This chapter deals with logarithms, log differentiation, This chapter deals with logarithms, log differentiation, log limits, and L’Hopital’s rule.log limits, and L’Hopital’s rule.
Seeing these words, you should know about exponents Seeing these words, you should know about exponents and logarithms. You also must remember derivatives, and logarithms. You also must remember derivatives, integrals, and limits. In effect, you are expected to do integrals, and limits. In effect, you are expected to do this chapter with a strong background of the previous this chapter with a strong background of the previous chapters. If you feel weak in any area, please take the chapters. If you feel weak in any area, please take the time to look at the previous chapters.time to look at the previous chapters.
You should also remember the log laws from pre-You should also remember the log laws from pre-calculuscalculus
LOG LAWS (from pre-LOG LAWS (from pre-calculus)calculus)
You should have remembered these lawsYou should have remembered these laws Multiplication: log(ab)=log a + log bMultiplication: log(ab)=log a + log b Division: log(a/b) = log a – log bDivision: log(a/b) = log a – log b Power: log (aPower: log (abb)= a log b)= a log b Conversion of base: logConversion of base: logaab= (ln b)/(ln a)b= (ln b)/(ln a)
EXPONENTIAL AND EXPONENTIAL AND LOGARITHMIC FUNCTIONSLOGARITHMIC FUNCTIONS
From a pre-calculus course, you should From a pre-calculus course, you should remember the typical exponential function y=aremember the typical exponential function y=axx, , where a>0 and x is any real number.where a>0 and x is any real number.
The inverse of the exponential function is the The inverse of the exponential function is the logarithm function. logarithm function.
This means: logThis means: logaa(a(axx) = a) = aloglogaaxx=x.=x. Logs with base 10 are called Logs with base 10 are called common logscommon logs. .
Denoted by “log x” i.e. log x = logDenoted by “log x” i.e. log x = log1010xx Logs with base “e” are called Logs with base “e” are called natural logsnatural logs. .
Denoted by “ln x” i.e. ln x = logDenoted by “ln x” i.e. ln x = logeexx
LOGARITHM AS AN LOGARITHM AS AN INTEGRALINTEGRAL
The logarithm can be The logarithm can be expressed as an integral expressed as an integral of 1/x. The integral of 1/x. The integral initially was defined as initially was defined as the area of the function the area of the function from 1 to t.from 1 to t.
By definition, if x=1, then By definition, if x=1, then the limits would be the the limits would be the same and therefore, same and therefore, ln(1) = 0.ln(1) = 0.
If x was 0, the function If x was 0, the function would be undefined, would be undefined, since 1/0 is undefined.since 1/0 is undefined.
If 0<x<1, then ln(x) If 0<x<1, then ln(x) would be negative.would be negative.
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DERIVATIVES OF DERIVATIVES OF EXPONENTIAL AND EXPONENTIAL AND LOGARITHMIC FUNCTIONSLOGARITHMIC FUNCTIONS The derivatives of the exponential functions The derivatives of the exponential functions
shouldn’t be that difficult. In fact, you should shouldn’t be that difficult. In fact, you should have had the “e” function and the “ln” functions have had the “e” function and the “ln” functions memorized. Nevertheless, I put them up memorized. Nevertheless, I put them up anyway.anyway. If you put “e” in for “a”, you will see If you put “e” in for “a”, you will see that it still works out!that it still works out!
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INTEGRALS OF INTEGRALS OF EXPONENTIAL AND EXPONENTIAL AND LOGARITHMIC FUNCTIONSLOGARITHMIC FUNCTIONS
With the exception of the “e” function, the integrals of the With the exception of the “e” function, the integrals of the logs and exponential functions wouldn’t be that easy. To logs and exponential functions wouldn’t be that easy. To derive the aderive the axx integral, you rewrite the function as e integral, you rewrite the function as ex*ln ax*ln a. If . If you use u-substitution to solve this integral, you will notice you use u-substitution to solve this integral, you will notice how u = x*ln a. If you differentiate, du/ln a = dx. Therefore, how u = x*ln a. If you differentiate, du/ln a = dx. Therefore, you will get the integral stated below.you will get the integral stated below.
The integral of ln x will be explained later in Chapter 8.The integral of ln x will be explained later in Chapter 8.
Ca
adxa
Cedxe
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lnC
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xxxdxx
Cxxxxdx
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ln
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lnln
LOGARITHMIC LOGARITHMIC DIFFERENTIATIONDIFFERENTIATION
It doesn’t seem as easy as it sounds. Yes, even though It doesn’t seem as easy as it sounds. Yes, even though derivative of ln (x) = 1/x.derivative of ln (x) = 1/x.
There will be some functions where you cannot There will be some functions where you cannot differentiate functions using the basic rules of differentiate functions using the basic rules of differentiation mentioned in Chapter 2 and 3.differentiation mentioned in Chapter 2 and 3.
Typically, such functions that require logarithmic Typically, such functions that require logarithmic differentiation are the following.differentiation are the following.
variablevariablevariablevariable.. Long product of functionsLong product of functions Very complicated mix of quotients and products.Very complicated mix of quotients and products.
EXAMPLEEXAMPLE
Differentiate the Differentiate the following function!following function!
Pretty tough since both Pretty tough since both x variables change.x variables change.
We can’t really use the We can’t really use the previous rules of previous rules of differentiation because differentiation because neither variable is neither variable is constant.constant.
xxy
LOGARITHMIC LOGARITHMIC DIFFERENTIATIONDIFFERENTIATION
Use natural logs to help Use natural logs to help you out.you out.
If you took the ln of xIf you took the ln of xxx, , then, you will get x*ln (x). then, you will get x*ln (x).
If you take the ln (y) and If you take the ln (y) and differentiate that differentiate that implicitlyimplicitly with respect to with respect to x, you will get (1/y) * x, you will get (1/y) * (dy/dx). (dy/dx).
Simply multiply by y and Simply multiply by y and you isolate the dy/dx and you isolate the dy/dx and thus, solved for the thus, solved for the derivative.derivative.
Don’t forget to substitute Don’t forget to substitute the y after multiplying!!!the y after multiplying!!!
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ln1
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ANALYZING LOGARITHMIC ANALYZING LOGARITHMIC DIFFERENTIATION DIFFERENTIATION
Notice how by the power of the natural Notice how by the power of the natural log function as well as the implicit log function as well as the implicit differentiation problem became easy.differentiation problem became easy.
Never forget the golden rule! Never forget the golden rule! ln(fln(fgg)=g*ln(f). <Given that f and g are )=g*ln(f). <Given that f and g are functions of x>functions of x>
Let’s try another one.Let’s try another one.
EXAMPLEEXAMPLE
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sincotln2
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sinlnlnln
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GivenGiven Take the logs of both Take the logs of both
sides and simplify.sides and simplify.
Differentiate implicitly Differentiate implicitly with respect to both with respect to both sides. Simplify if sides. Simplify if necessary.necessary.
Multiply both sides by y.Multiply both sides by y. Substitute.Substitute. Leave your answer like Leave your answer like
this! Don’t bother this! Don’t bother distributing!distributing!
LOGARITHMIC LOGARITHMIC DIFFERENTIATIONDIFFERENTIATION
Amazing how a innocent looking function Amazing how a innocent looking function has a monstrous answer.has a monstrous answer.
Also amazing how a difficult function can Also amazing how a difficult function can be broken down by logarithms.be broken down by logarithms.
Always remember to simplify as much as Always remember to simplify as much as possible.possible.
LIMITSLIMITS
Remember Chapter 1? The prelude to Remember Chapter 1? The prelude to calculus? The one and the only, the limit!calculus? The one and the only, the limit!
If functions could be monstrous for If functions could be monstrous for differentiating, they can always be differentiating, they can always be monsters for taking the limit, since the monsters for taking the limit, since the laws of limits were straight-forward.laws of limits were straight-forward.
0/0 and ∞/∞0/0 and ∞/∞
Remember at times in Chapter 1, there Remember at times in Chapter 1, there have been cases where in quotients have been cases where in quotients there would be a zero in the denominator there would be a zero in the denominator or 0/0 or something like ∞/∞ ?or 0/0 or something like ∞/∞ ?
Now, there is a special way how to attack Now, there is a special way how to attack these limits without having to surrender these limits without having to surrender to algebra too many times.to algebra too many times.
L’HOPITAL’S RULE!L’HOPITAL’S RULE!
You can solve this problem by the use of the You can solve this problem by the use of the one and only……one and only……
L’HOPITAL’S RULE!L’HOPITAL’S RULE! Basically, it says if you have a rational function Basically, it says if you have a rational function
h(x) = f(x)/g(x) and if the limit of the f(x) and h(x) = f(x)/g(x) and if the limit of the f(x) and g(x) both are 0 or both ∞, then you can simply g(x) both are 0 or both ∞, then you can simply take the derivative of f(x) and g(x) and apply take the derivative of f(x) and g(x) and apply limit again.limit again.
In other words, the limit of f(x)/g(x) is the same In other words, the limit of f(x)/g(x) is the same as the limit of f’(x)/g’(x)as the limit of f’(x)/g’(x)
L’HOPITAL’S RULEL’HOPITAL’S RULE
Simply put…Simply put…
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EXAMPLE:EXAMPLE:
Given the old classical Given the old classical problem…problem…
If you differentiate the If you differentiate the top function, you get 2x. top function, you get 2x. If you differentiate the If you differentiate the bottom function you get bottom function you get 1.1.
Taking the limit of 2x as Taking the limit of 2x as x approaches 5 is a x approaches 5 is a piece of cake.piece of cake. 10)5(2
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25lim
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xxx
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ANOTHER EXAMPLEANOTHER EXAMPLE
Given the rational Given the rational function.function.
An asymptote is An asymptote is predicted since they both predicted since they both are of equal degree. It is are of equal degree. It is obviously 1, but let’s obviously 1, but let’s prove it using L’Hopital’s prove it using L’Hopital’s rule.rule.
Always remember, if you Always remember, if you are stuck with 0/0 or are stuck with 0/0 or ∞/∞, then always use ∞/∞, then always use L’Hopital’s rule until you L’Hopital’s rule until you can actually compute the can actually compute the limit!limit!
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23lim
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CONDITION FOR CONDITION FOR L’HOPITAL’S RULEL’HOPITAL’S RULE
It must be a RATIONAL FUNCTIONIt must be a RATIONAL FUNCTION The limit be in a form of ∞/∞ or 0/0, The limit be in a form of ∞/∞ or 0/0,
before applying the derivative.before applying the derivative. If the limit was able to be computed, yet If the limit was able to be computed, yet
you differentiate again, you might not the you differentiate again, you might not the right limit.right limit.
LIMITS BY LOGARITHMSLIMITS BY LOGARITHMS
If we can differentiate using logarithms, If we can differentiate using logarithms, you can certainly take the limits using you can certainly take the limits using logs.logs.
Same rules apply for taking limits of logs. Same rules apply for taking limits of logs. Use whatever log rules, algebra, and limit Use whatever log rules, algebra, and limit laws possiblelaws possible
FIND THE LIMIT OF THIS FIND THE LIMIT OF THIS FUNCTIONFUNCTION
Find the following limit.Find the following limit. Tough using the laws of limits. If 1/x = 1/Tough using the laws of limits. If 1/x = 1/∞∞ =0, then 1+0 =0, then 1+0
= 1. However 1= 1. However 1∞∞ =1 not necessarily define the proper =1 not necessarily define the proper limit, since limit, since ∞∞ is such a vague number. is such a vague number.
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x x
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LIMITS BY LOGSLIMITS BY LOGS
Let the function inside the Let the function inside the limit equal y.limit equal y.
Things will look a little more Things will look a little more simpler once you use the ln simpler once you use the ln function.function.
PROBLEM!! The limit of x will PROBLEM!! The limit of x will be be ∞, while the limit of the of ∞, while the limit of the of the ln function will be 0. If the the ln function will be 0. If the ∞ was in the denominator, it ∞ was in the denominator, it would become a 0. Therefore, would become a 0. Therefore, you got a 0/0 scenario.you got a 0/0 scenario.
Time to use L’Hopital’s rule!Time to use L’Hopital’s rule!
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LIMITS BY LOGSLIMITS BY LOGS
You can rewrite x, as You can rewrite x, as 1/(1/x). That is to say, 1/(1/x). That is to say, put 1/x in the put 1/x in the denominator. Now, it’s a denominator. Now, it’s a rational function which rational function which L’Hopital’s Rule applies.L’Hopital’s Rule applies.
Now it is possible to Now it is possible to differentiate the top and differentiate the top and bottom.bottom.
Simply and apply the Simply and apply the limit.limit.
The limit is simply 1.The limit is simply 1.
11limlnlim
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1limlnlim
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limlnlim
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DON’T FORGET THOUGH!DON’T FORGET THOUGH! The limit of the ln(y) = 1.The limit of the ln(y) = 1. We must find the limit of y, not We must find the limit of y, not
ln y.ln y. Remember limit of a function is Remember limit of a function is
the same as performing the the same as performing the function on the limit.function on the limit.
Since the you got a ln on one Since the you got a ln on one side and a constant on the side and a constant on the other side, you can simply other side, you can simply exponentiate both sides. exponentiate both sides. Remember that ln and e are Remember that ln and e are inverses.inverses.
So this special limit yields the So this special limit yields the number known as the backbone number known as the backbone of calculus! “e”!!of calculus! “e”!!
ex
ey
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11lim
lim
1limlnlnlim
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IMPORTANCE OF IMPORTANCE OF LOGARITHMSLOGARITHMS
As you have seen throughout this chapter, the As you have seen throughout this chapter, the concept of logs have been proven quite concept of logs have been proven quite important in order to do simple calculus important in order to do simple calculus functions like limits and intermediate functions functions like limits and intermediate functions like differentiation and integration.like differentiation and integration.
In addition, “e” and “ln” are the backbone of In addition, “e” and “ln” are the backbone of calculus. You cannot do anything without these calculus. You cannot do anything without these two powerful functions in calculus.two powerful functions in calculus.
SUMMARYSUMMARY
You must know how to differentiate and You must know how to differentiate and integrate the exponential and logarithmic integrate the exponential and logarithmic functions.functions.
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SUMMARYSUMMARY
When you are differentiating monstrous looking When you are differentiating monstrous looking functions, or functions with variables in the functions, or functions with variables in the exponent and the base, both, then use exponent and the base, both, then use logarithmic differentiation! Don’t forget that you logarithmic differentiation! Don’t forget that you are differentiating implicitly with respect to x.are differentiating implicitly with respect to x.
When you have ∞/∞ or 0/0 in a rational When you have ∞/∞ or 0/0 in a rational function, use L’Hopital’s rule. If f(x)/g(x) function, use L’Hopital’s rule. If f(x)/g(x) produces undefined results, keep produces undefined results, keep differentiating f(x) and g(x) until you can differentiating f(x) and g(x) until you can evaluate the limit.evaluate the limit.
When evaluating complicated limits, use When evaluating complicated limits, use logarithms to make things look easier.logarithms to make things look easier.
END NOTESEND NOTES
Originally, this chapter was about “transcendental Originally, this chapter was about “transcendental functions.” Exponential and logarithmic functions are functions.” Exponential and logarithmic functions are certainly “transcendental functions.” Originally, this certainly “transcendental functions.” Originally, this chapter included hyperbolic functions. However, due to chapter included hyperbolic functions. However, due to the growing number of curricula that don’t teach the growing number of curricula that don’t teach hyperbolic functions anymore, I have chose not include hyperbolic functions anymore, I have chose not include it.it.
Since this chapter included a great deal of logarithms Since this chapter included a great deal of logarithms and exponential functions, I renamed this chapter.and exponential functions, I renamed this chapter.
The following chapter is about how to integrate The following chapter is about how to integrate functions in general. You must know all the algebra, functions in general. You must know all the algebra, and the calculus learned so far. If not, and the calculus learned so far. If not, REVIEWREVIEW chapters 1 -7! You may not have a second chance chapters 1 -7! You may not have a second chance after this pointafter this point
END OF CHAPTER SEVENEND OF CHAPTER SEVEN
jaya sri krsna caitanya prabhu nityanandajaya sri krsna caitanya prabhu nityananda sri advaita gadadhara sri vasadi gaura bhakta sri advaita gadadhara sri vasadi gaura bhakta
vrndavrnda hare krsna hare krsna krsna krsna hare harehare krsna hare krsna krsna krsna hare hare hare rama hare rama rama rama hare harehare rama hare rama rama rama hare hare
CREDITSCREDITS
Dr. A. MoslowDr. A. Moslow Mr. G. ChomiakMr. G. Chomiak Mr. J. TrapaniMr. J. Trapani Single-Variable Calculus (SUNY Buffalo)Single-Variable Calculus (SUNY Buffalo) Calculus and Early Transcendental Calculus and Early Transcendental
Functions 5Functions 5thth Ed. Ed. ““Dr. Math” for the logarithmic limits Dr. Math” for the logarithmic limits
question on the limit resulting “e.”question on the limit resulting “e.”
