Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts

Saturday, November 17, 2012

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.)


Expand.
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.
Evaluate
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).
Simplify.
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Friday, November 16, 2012

Exponential Functions and Their Graphs

Up to this point we have limited our study to constant exponents.  Now we will explore functions with variable exponents.
Here are some examples:
Evaluate.
When asked to graph these types of functions we plot points.

   Step 1: Choose any values for x. Choose some negative values, zero, and some positive values.
   Step 2: Evaluate to find the corresponding y-values.
   Step 3: Plot the points. The points you plot the better the graph will look.
 
Notice that the y-values can never actually attain zero.  They get infinitely close when x is negative so the graph will be asymptotic to the x-axis.  In other words, the line y = 0 is a horizontal asymptote.

Instructional Video: Graph Exponential Functions

 
Looking at the two previously graphed exponential functions on the same set of axes we will be able compare their rates of growth.
Notice that for positive values of x, g(x) increases more rapidly than f(x). Also the domain and range for the two functions are the same. In addition, they are both asymptotic to the x-axis (or y = 0) and they both have the same y-intercept (0, 1). Now we will explore exponential functions with bases greater than zero but less than 1.

 
 

Tip: All exponential functions of the form y = b^have (0, 1) as a y-intercept, no x-intercept, and the x-axis will be a horizontal asymptote.

Looking at the two exponential two previous exponential functions on the same set of axes we will be able compare their rates of growth.
Notice that g(x) decreases more rapidly than f(x). Also the domain, range, and y-intercept for the two functions are the same.
 
 
Tip: All exponential functions of the form y = b^(x+h) + k  have a horizontal asymptote at y = k.
It is also useful to note the rigid transformations in the above graphs.  In the previous example the basic graph of y = 3^x was shifted down two units and to the left 1 unit.
 
 
 

Notice that all of the exponential functions graphed above pass the horizontal line test.  Therefore we may conclude that they are all one-to-one and have an inverse.  This is one of the most important observations of this section. We will define the inverse of these functions in subsequent sections.

A common error when stating the range is to include the y-value that defines the asymptote.  In the previous problem one might think the range is [ 2, inf ) but this is incorrect.  The y-values get infinitely close to 2 but never actually attain that value – this is what it means to be asymptotic.  So the range is ( 2, inf ).

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