- Just a note that there is now a link to section summaries off of the homepage.
- Your first assignment is due today.
- Logs are, by definition, the inverses of the exponential functions:
- The laws of logarithms are reflections of the properties of
exponentials:
- The derivation of the derivative of the natural log function is
easy, given our work on inverses:
We can demonstrate this easily given our theorem about the derivatives of inverses.
- Another way to define the log function is as the anti-derivative
of the "weird power"
:
Symmetry help us to understand the absolute value in this result: an antiderivative of the odd function 1/x is an even function ln(|x|).
This is a very general and beautiful result, which you should think about carefully.
For those who don't remember:
- An even function satisfies
- An odd function satisfies
- An even function satisfies
- Examples:
- #7, p. 408 (started this one last time)
- 10
- 26
- 44
- 67
- 73
- We want to spend a minute recognizing that we can turn any
exponential or logarithm of base b into another with base
e:
- We also want to remember that the natural log, ln(x), as
the inverse of an increasing function,
, is also increasing; but, using the mirror, the increase in the exponential function, being extremely rapid, is reflected as an extremely slow growth in the natural logarithm.
- This section features the logarithm taking care of some tricky cases that we couldn't handle previously, through a process called "logarithmic differentiation".
Let's see this process in action, on an interesting function: the example 16, p. 417.
- Let's look at the summary section 6.4 for some of the results of this section.
Now I'd like to go over the result following example 16, entitled "The Number e as a Limit". It incorporates some nice elements of limits, including that limit definition.