Calculus I, Pre-Lab 9: The Logarithmic Function as an Inverse Function

Name:

The Pre-lab is intended to encourage you to prepare for the Lab, so answer these questions in your own words, and hand this sheet in at the beginning of lab.

  1. Read activity #1 in Lab 9 and section 3.2 in the text. Now answer the following questions.
  1. What does it mean for a function to be one-to-one?
  2. Why is it necessary that a function be one-to-one in order for it to possess an inverse?
  3. In EXAMPLE 5 on the bottom of page 206 why is the function described by the equation , not one-to-one but the function described by the equation , with one-to-one?
  1. In activity #2 in Lab 9 you meet the inverse to the exponential function which we call the natural logarithm function and we might write . We can describe the exponential function f in words by saying: "exp is the process that accepts a real number and returns the number e raised to that real number."
  1. Describe in words its inverse, the natural logarithm function.
  2. Find in Section 3.3 the symbolic representation of the following statements

STATEMENT / SYMBOLIC TRANSLATION
When we divide two numbers written in exponential form that have like bases we subtract their exponents.
When we multiply two numbers written in exponential form that have like bases we add their exponents.
When we raise a number written in exponential form to another number we multiply the exponents.
  1. You have no doubt encountered this natural logarithm in your mathematical travels before (maybe like you encounter a dear friend, or maybe like you encounter the friend who is always hitting you up for a dime.) Look at FIGURE 3 on page 212. We would like you to think about how slowly the graph of is growing. Begin by filling in the following table of values.

X / / x /
e-3 / e16
e0 / e20
e8 / e30

Now suppose you are going to sketch a graph of the logarithm function that includes the six ordered pairs determined above. You are going to let 1/3 of an inch represent 1 unit in both the vertical and horizontal direction. How wide will the graph be if it is to be only 11 inches high? (Answer in miles!)

Is the function a decreasing or increasing function?

So what does this say about the sign of its derivative?

Is the derivative of the logarithmic function ever 0?

Compute from the table above the slope of the secant line joining the points (e16, 16) and (e20, 20).

Is this slope an over or under approximation to f ‘ (e20)? (Think about the way the graph "bends".)

  1. Read section 3.5 and come up with two good reasons why we study the exponential function.

Reason 1:
Reason 2: