THE CALCULUS OF LOGARITHMIC FUNCTIONS THE CALCULUS OF LOGARITHMIC FUNCTIONS

PREREQUISITES  This chapter is generally the start of the Calculus II curriculum.  This chapter deals with logarithms, log differentiation, log limits, and L’Hopital’s rule.  Seeing these words, you should know about exponents and logarithms. You also must remember derivatives, integrals, and limits. In effect, you are expected to do this chapter with a strong background of the previous chapters. If you feel weak in any area, please take the time to look at the previous chapters.  You should also remember the log laws from pre- calculus

LOG LAWS (from pre- calculus)  You should have remembered these laws  Multiplication: log(ab)=log a + log b  Division: log(a/b) = log a – log b  Power: log (a b )= a log b  Conversion of base: log a b= (ln b)/(ln a)

EXPONENTIAL AND LOGARITHMIC FUNCTIONS  From a pre-calculus course, you should remember the typical exponential function y=a x, where a>0 and x is any real number.  The inverse of the exponential function is the logarithm function.  This means: log a (a x ) = a log a x =x.  Logs with base 10 are called common logs. Denoted by “log x” i.e. log x = log 10 x  Logs with base “e” are called natural logs. Denoted by “ln x” i.e. ln x = log e x

LOGARITHM AS AN INTEGRAL  The logarithm can be expressed as an integral of 1/x. The integral initially was defined as the area of the function from 1 to t.  By definition, if x=1, then the limits would be the same and therefore, ln(1) = 0.  If x was 0, the function would be undefined, since 1/0 is undefined.  If 0<x<1, then ln(x) would be negative.

DERIVATIVES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS  The derivatives of the exponential functions shouldn’t be that difficult. In fact, you should have had the “e” function and the “ln” functions memorized. Nevertheless, I put them up anyway. If you put “e” in for “a”, you will see that it still works out!

INTEGRALS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS  With the exception of the “e” function, the integrals of the logs and exponential functions wouldn’t be that easy. To derive the a x integral, you rewrite the function as e x*ln a. If you use u-substitution to solve this integral, you will notice how u = x*ln a. If you differentiate, du/ln a = dx. Therefore, you will get the integral stated below.  The integral of ln x will be explained later in Chapter 8.

LOGARITHMIC DIFFERENTIATION  It doesn’t seem as easy as it sounds. Yes, even though derivative of ln (x) = 1/x.  There will be some functions where you cannot differentiate functions using the basic rules of differentiation mentioned in Chapter 2 and 3.  Typically, such functions that require logarithmic differentiation are the following.  variable variable.  Long product of functions  Very complicated mix of quotients and products.

EXAMPLE  Differentiate the following function!  Pretty tough since both x variables change.  We can’t really use the previous rules of differentiation because neither variable is constant.

LOGARITHMIC DIFFERENTIATION  Use natural logs to help you out.  If you took the ln of x x, then, you will get x*ln (x).  If you take the ln (y) and differentiate that implicitly with respect to x, you will get (1/y) * (dy/dx).  Simply multiply by y and you isolate the dy/dx and thus, solved for the derivative.  Don’t forget to substitute the y after multiplying!!!

ANALYZING LOGARITHMIC DIFFERENTIATION  Notice how by the power of the natural log function as well as the implicit differentiation problem became easy.  Never forget the golden rule! ln(f g )=g*ln(f).  Never forget the golden rule! ln(f g )=g*ln(f).  Let’s try another one.

EXAMPLE  Given  Take the logs of both sides and simplify.  Differentiate implicitly with respect to both sides. Simplify if necessary.  Multiply both sides by y.  Substitute.  Leave your answer like this! Don’t bother distributing!

LOGARITHMIC DIFFERENTIATION  Amazing how a innocent looking function has a monstrous answer.  Also amazing how a difficult function can be broken down by logarithms.  Always remember to simplify as much as possible.

LIMITS  Remember Chapter 1? The prelude to calculus? The one and the only, the limit!  If functions could be monstrous for differentiating, they can always be monsters for taking the limit, since the laws of limits were straight-forward.

0/0 and ∞/∞  Remember at times in Chapter 1, there have been cases where in quotients there would be a zero in the denominator or 0/0 or something like ∞/∞ ?  Now, there is a special way how to attack these limits without having to surrender to algebra too many times.

L’HOPITAL’S RULE!  You can solve this problem by the use of the one and only……  L’HOPITAL’S RULE!  Basically, it says if you have a rational function 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 take the derivative of f(x) and g(x) and apply limit again.  In other words, the limit of f(x)/g(x) is the same as the limit of f’(x)/g’(x)

L’HOPITAL’S RULE  Simply put…

EXAMPLE:  Given the old classical problem…  If you differentiate the top function, you get 2x. If you differentiate the bottom function you get 1.  Taking the limit of 2x as x approaches 5 is a piece of cake.

ANOTHER EXAMPLE  Given the rational function.  An asymptote is predicted since they both are of equal degree. It is obviously 1, but let’s prove it using L’Hopital’s rule.  Always remember, if you are stuck with 0/0 or ∞/∞, then always use L’Hopital’s rule until you can actually compute the limit!

CONDITION FOR L’HOPITAL’S RULE  It must be a RATIONAL FUNCTION  The limit be in a form of ∞/∞ or 0/0, before applying the derivative.  If the limit was able to be computed, yet you differentiate again, you might not the right limit.

LIMITS BY LOGARITHMS  If we can differentiate using logarithms, you can certainly take the limits using logs.  Same rules apply for taking limits of logs. Use whatever log rules, algebra, and limit laws possible

FIND THE LIMIT OF THIS FUNCTION  Find the following limit.  Tough using the laws of limits. If 1/x = 1/∞ =0, then 1+0 = 1. However 1 ∞ =1 not necessarily define the proper limit, since ∞ is such a vague number.

LIMITS BY LOGS  Let the function inside the limit equal y.  Things will look a little more simpler once you use the ln function.  PROBLEM!! The limit of x will be ∞, while the limit of the of the ln function will be 0. If the ∞ was in the denominator, it would become a 0. Therefore, you got a 0/0 scenario.  Time to use L’Hopital’s rule!

LIMITS BY LOGS  You can rewrite x, as 1/(1/x). That is to say, put 1/x in the denominator. Now, it’s a rational function which L’Hopital’s Rule applies.  Now it is possible to differentiate the top and bottom.  Simply and apply the limit.  The limit is simply 1.

DON’T FORGET THOUGH!  The limit of the ln(y) = 1.  We must find the limit of y, not ln y.  Remember limit of a function is the same as performing the function on the limit.  Since the you got a ln on one side and a constant on the other side, you can simply exponentiate both sides. Remember that ln and e are inverses.  So this special limit yields the number known as the backbone of calculus! “e”!!

IMPORTANCE OF LOGARITHMS  As you have seen throughout this chapter, the concept of logs have been proven quite important in order to do simple calculus functions like limits and intermediate functions like differentiation and integration.  In addition, “e” and “ln” are the backbone of calculus. You cannot do anything without these two powerful functions in calculus.

SUMMARY  You must know how to differentiate and integrate the exponential and logarithmic functions.

SUMMARY  When you are differentiating monstrous looking functions, or functions with variables in the exponent and the base, both, then use logarithmic differentiation! Don’t forget that you are differentiating implicitly with respect to x.  When you have ∞/∞ or 0/0 in a rational function, use L’Hopital’s rule. If f(x)/g(x) produces undefined results, keep differentiating f(x) and g(x) until you can evaluate the limit.  When evaluating complicated limits, use logarithms to make things look easier.

END NOTES  Originally, this chapter was about “transcendental functions.” Exponential and logarithmic functions are certainly “transcendental functions.” Originally, this chapter included hyperbolic functions. However, due to the growing number of curricula that don’t teach hyperbolic functions anymore, I have chose not include it.  Since this chapter included a great deal of logarithms and exponential functions, I renamed this chapter.  The following chapter is about how to integrate functions in general. You must know all the algebra, and the calculus learned so far. If not, REVIEW chapters 1 -7! You may not have a second chance after this point

END OF CHAPTER SEVEN  jaya sri krsna caitanya prabhu nityananda  sri advaita gadadhara sri vasadi gaura bhakta vrnda  hare krsna hare krsna krsna krsna hare hare  hare rama hare rama rama rama hare hare

CREDITS  Dr. A. Moslow  Mr. G. Chomiak  Mr. J. Trapani  Single-Variable Calculus (SUNY Buffalo)  Calculus and Early Transcendental Functions 5 th Ed.  “Dr. Math” for the logarithmic limits question on the limit resulting “e.”

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