Exponential and Logarithmic Functions (1/30/09) The formula of the general exponential function P (t) of a variable t is P (t) = P 0 a t where P 0 is the value when t = 0 (the “initial value”) and a, a positive number, is called the base. Often the number e (  2.718) is used as the base: P (t) = P 0 e kt where k is the “continuous growth rate.”

Clicker Question 1 What is the value of P (t ) = 100(2 t ) when t = 0 and when t = 7? A. 0 and 1400 B. 100 and 1400 C. 0 and D. 100 and E. 200 and 12800

New Functions from Old: Inverse Functions Inverse functions: If a function can be reversed (i.e., the output becomes the input and the input becomes the output), the reverse function is called the inverse of the original. For example: the inverse of f (x) = x 3 is the cube root function. A function is (directly) invertible only if its graph hits every horizontal line at most once (it is “one-to-one”).

Inverses - Examples f (x ) = x 2 is not directly invertible. (Why?) But if we restrict its domain so that it is one-to-one, then we can invert it on that domain. (To what?) In general to invert a given formula: Write in the form y = f (x ) Solve for x. Exchange x and y in your formula Examples….

Clicker Question 2 Find a formula (y as a function of x ) for the inverse of y = (2x +1) 3. A. y = (2x +1) 1/3 B. y = (2x +1) -3 C. y = (x 1/3 – 2) + 1 D. y = (x 1/3 – 1) / 2 E. y = 2x 1/3 - 1

Inverse Trig Functions The sin and cos functions cannot be directly inverted. (Why?) But restricting the domains appropriately (How? Look at the graphs!), we obtain the inverse, or arc, trig functions, arcsin, arccos, arctan, etc. These functions take in a number and put out an angle. So, e.g., arcsin(x ) means “the angle whose sin is x.”

Clicker Question 3 What is the arcsin(-  2 / 2)? A. 45 ° B. -  / 4 C.  / 4 D ° E. 0

Logarithmic Functions Logarithmic functions (“log” functions) are the inverses of exponential functions. That is, an exponential function has as input an exponent on a fixed base, whereas a log function has the exponent on a fixed base as output !! For example, the log function log 10 (x) has as output the exponent on 10 to give you x. Hence, for example, log 10 (1000) = 3.

Clicker Question 4 What is log 3 (1 / 81)? A. – 4 B. 4 C D E. I’m allergic to log functions

Properties of Log Functions Since the output of logs are exponents, all log functions: Turn products into sums, i.e., log(A B) = log(A) + log(B) Turn quotients into differences, i.e., log(A / B) = log(A) – log(B) Turn exponents into coefficients, i.e., log(A p ) = p log(A)

Bases of Exp & Log Functions Any positive number can be used as the base for log (and exponential) functions (1, however, is not a useful base….) The two most common bases are 10 and e (which is about ). Base 2 is also quite common (especially in computer science). Log e is commonly denoted ln and is called the “natural log function”. It is used in calculus (along with y = e x ) because there it’s the easiest!

Using Logs to Solve Equations Because logs are the inverses of exponential functions, they are used to solve any equation in which the unknown is an exponent. Simply take the log (any base you wish) of both sides and use the property of logs that they turn exponents into coefficients. Example: If 5 x = 15, then ln(5 x ) = ln(15), so x ln(5) = ln(15), so x = ln(15) / ln(5) = 1.683

Clicker Question 5 Solve the equation 4 t = 14 to 3 decimal places. A B C D E

About that number e …. What is this mysterious number e anyway? Well, it is the natural limit of compounding, in the following sense: If you invest $1 for 1 year at 100% interest compounded annually, you make $1(1+1) 1 = $2.00 If you compound twice, you make $1(1 + ½) 2 = $2.25

The story of e, continued Compound quarterly: $1(1+1/4) 4 = $2.44 Compound monthly: $1(1+1/12) 12 = $2.61 Compound daily: $1(1+1/365) 365 = $2.71 Definition: e = limit (1+1/n) n as n gets big. So, compound continuously, make $e !!! We will see that e is “natural” in various ways as this course goes on.

Assignment Monday we will have Lab #1 in Harder 209. You will be expected to pair up with a partner for this (and every) lab. You do not need your clicker or your text for labs unless I tell you otherwise. For Wednesday: Read Sections 1.5 and 1.6. In Section 1.5 do Exercises 5, 9, 15, 19, 21, and 25. In Section 1.6 do Exercises 1, 2, 3-13 odd, odd, odd, odd, odd.