# In doing the exercises related to this section of the

This preview shows pages 2–3. Sign up to view the full content.

In doing the exercises related to this section of the module (and in general), you will need to use the “ y x ” or “ x y ” button on your scientific calculator. Typically, the sequence of commands for, say, “2 3 = ?” is [2 x y 3 =], but if you haven’t used this button before, you should experiment with some simple examples, before doing the exercises. Logarithms and Exponential Functions: If b x = a , then x is the logarithm of a to base b , that is, the power to which b must be raised to give a : log b a = x. The two most common bases for logarithms are 10 and e . Logarithms to these two bases are written “log” (or log 10 ) and ln (for “natural” logarithm), respectively. Recall that e is a transcendental number (with a value of approximately 2.71828) defined as the limit of the expression (1 + 1/ n ) n as n approaches infinity, a formula which represents instantaneous compounding of a growth rate, as n , the number of subperiods within a period, becomes infinite and the growth rate per subperiod becomes infinitesimal. It is possible to convert logarithms from one base to another. If b x = a y , then x log b b = y log b a , and since the logarithm of any number to itself as base (in this case, log b b ) = 1, we have x = y log b a . Your scientific calculator has log 10 , 10 x , ln and e x functions. You may want to confirm by punching 10 into your calculator and hitting the ln button that log e 10 = ln 10 = approximately 2.302585. Hence if log 10 a = x, then log e a = 2.302585 x . Using our knowledge of exponents, we know immediately that log 10 10 = 1, log 10 100 = 2, log 10 1 = 0, and log 10 0.1 = log 10 1/10 = –1. We also know that numbers between 10 and 100 have logarithms between 1 and 2, and we trust our calculators to remember precisely what they are. We shall mention only 2 applications of logarithms and exponents here. First, using a result from Module 10, we have that for the function ln x , the derivative d ( ln x )/ dx = 1/ x, and so d ( ln x ) = dx/x . Since the same relation holds true for a variable

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern