Showing posts with label common logarithm. Show all posts
Showing posts with label common logarithm. Show all posts

Properties of the Logarithm

The following properties of the logarithm are derived from the rules of exponents.

The properties that follow below are derived from the fact that the logarithm is defined as the inverse of the corresponding exponential.

To familiarize ourselves with the properties we will first expand the following logarithms. (Assume all variables are positive.)

Notice that there is no explicit property that allows us to work with nth roots within the argument of the logarithm.  To simplify these, first change the root to a rational exponent then apply the power rule.
When expanding, notice that we must use the same base throughout the expression. For the next set of problems we will first use the properties to expand then substitute in the appropriate values as the last step.
Expanding is useful for learning the rules and properties associated with logarithms but as it turns out, in practice, condensing down to a single logarithm is the skill that we really need to focus on.

Rewrite as a single logarithm (condense).
Tip: When simplifying these down to one logarithm use only one operation at a time and work from left to right. Combining or skipping steps usually leads to mistakes. Do not race, work slowly, and pay attention to notation.

Evaluate (Round to the nearest ten thousandths where appropriate).

YouTube Videos:

Logarithmic Functions

Logarithm Playlist on YouTube

In the last section we made note of the fact that exponential functions pass the horizontal line test and thus have an inverse.

YouTube Playlist: Logarithms and their Graphs

 Using the steps for finding the inverse we obtain:
At this point there is no way to solve for y. Therefore we make the following definition:
Here are some examples of logarithmic facts and their equivalent in exponential form.
Use this definition to rewrite the following in logarithmic form.

Use the proper terminology when reading logarithms,
reads “log base 5 of 125 is 3.”  If given
then x is called the argument of the logarithm. Also, notice that y is an exponent so the logarithm is actually an exponent, this will be important later.

Evaluate without using a calculator.

Instructional Video: Introduction to Logarithms

The base b can be any real number greater than zero but not including one - 10 and the natural base e are used often.
A logarithm without a base is interpreted as the common logarithm.

Often the logarithms do not work out so nicely and we will need to use a calculator to evaluate them.
Here are the steps for using a TI-30x calculator.  Other scientific calculators are similar.
So if we round off the nearest thousandth we have
Evaluate using a calculator rounding off to the nearest thousandth.
YouTube Videos: