{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

In doing the exercises related to this section of the

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

View Full Document Right Arrow Icon
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
Background image of page 2

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

View Full Document Right Arrow Icon
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}