## Algebra

Showing posts with label exponential. Show all posts
Showing posts with label exponential. Show all posts

### Solving Exponential Equations

In this section, we will make use of what we have learned about exponential functions to solve equations.

Make use of the above property if you are able to express both sides of the equation in terms of the same base.

Step 1: Express both sides in terms of the same base.
Step 2: Equate the exponents.
Step 3: Solve the resulting equation.

Solve.
It is not always the case that we will be able to express both sides of an equation in terms of the same base.  For this reason we will make use of the following property.
Make use of the above property if you are unable to express both sides of the equation in terms of the same base.

Step 1: Isolate the exponential and then apply the logarithm to both sides.
Step 2: Apply the power rule for logarithms and write the exponent as a factor of the base.
Step 3: Solve the resulting equation.
Solve.

When solving exponential equations and using the above process, the rule of thumb is to choose the common logarithm unless the equation involves the natural base e.   We choose these because there is a button for them on the calculator. However, we could certainly choose any base that we wish; this is the basis for the derivation of the change of base formula.

YouTube Videos:

### The Natural Exponential Function

In this section we will introduce the natural base and define and graph the natural exponential function.
YouTube playlist for steps covering exponential functions and their graphs
This number is so special, just as is pi, that it has its own button on your scientific calculator. Try finding the e^x button on your calculator and using it to calculate e.

Evaluate.

Graph the following exponential functions. State the domain, range, and label the horizontal asymptote.

YouTube Videos:

### 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 ).

YouTube Videos: