Wednesday 26 March class

Topics:

• Basic exponential functions and their graphs (for b>1 and for 0<b<1): See Example 13.2 and Observation 13.3, also Example 13.5

Note that all of these basic graphs have the y-intercept at (0,1) and have a horizontal asymptote y=0 (the x-axis). The graph is asymptotic to the x-axis on only one side, though. The domain is  and the range is the interval . (The graph always lies above the x-axis.)

 

• The Euler number , also known as the base of the natural logarithm. See Definition 13.4.

 

• Transformations of the basic exponential graphs: see Example 13.7 (I changed (c) though)

 

• Definition of the logarithm as the inverse of a basic exponential function: For any base b>o, b not equal to 1, we have

 

• Special cases:

When the base is 10, we call it the common logarithm, and we write it

When the base is , we call it the natural logarithm, and we write it .

(However, be aware that many mathematicians use  to mean the natural logarithm, because the natural logarithm is by far the most commonly used logarithm for us!)

 

• Graphs of basic logarithmic functions with b>1. See Example 13.12.

Note that these graphs have x-intercept at (1,0) and have a vertical asymptote x=0 (the y-axis). The domain is the interval and the range is . [Remember that the inverse of a function interchanges the roles of the inputs and outputs: compare these to what we had for the basic exponential graphs!]

 

• Computing some values of logarithms using the definition to rewrite into exponential form: see Example 13.10.

 

A few other things that came up during class today (while discussing the homework): I am putting them below the fold, at the bottom of this post.

Homework:

• Review all the definitions and examples worked in class, also the additional examples listed above in the textbook.

• Do the assigned problems from Session 13, except SKIP #13.2(d) and #13.6 (for now)

• Do the WeBWorK: start early, and make sure that you have an email address in WeBWorK (look under “password/email” in the left sidebar in WeBWorK). This WeBWorK is due by 11 PM Sunday.

• Do the Warm-Up for Properties of Logarithms: also due by 11 PM Sunday!

A few other things that came up during class today below the fold…

• Make sure that you know the difference of squares pattern:

• Some third-degree polynomials can be factored using the factoring by grouping method: this can be a very useful thing to know! The student who did Exercise 11.4(d) for us used this method to factor its numerator:

factor out the GCF of the first two terms, and factor out the GCF of the last two terms: (when the first term of a pair has a negative coefficient, we factor out a negative number in the GCF)

Now factor out the common factor (x-3):

We can factor this further by noticing that is a difference of squares:

Notice that the grouping method only works if the binomial factor (which was x-3 in this example) is the same both places when you factor out the GCF of each pair of terms. If that factor is not the same, the grouping method cannot help you.

 

For example, we also solved Exercise 12.2(h) where we had to factor

If you try to use grouping in this example, it will not work:

Notice that the binomial factors in parentheses are not the same, so we cannot go any further. This polynomial cannot be factored by grouping: we have to use the tools from Session 10 to factor it.

 

• Remember that when you are solving polynomial inequalities, the second step (after you make one side 0) is to solve the related equation. There is no way to solve a polynomial inequality by factoring the polynomial and leaving it as an inequality! This is a very common student error: don’t fall into that trap!

• Also beware of incorrect canceling: you cannot cancel the in the numerator and denominator of the rational expression

(Why not?)