Unit 7 exponential and logarithmic functions homework 9 answers

7a. evaluate exponential and logarithmic functions

Compare the rate of change of a linear function to an exponential function.
Define an exponential function. What are possible values for the base of an exponential function?
Explain how to evaluate an exponential function with base
.
Define a logarithmic function.
Compare common logarithms to natural logarithms.
Compare evaluating exponential functions to evaluating logarithmic functions.

An exponential function is a function of the form , where and are positive real numbers and ; the coefficient a represents the starting value and the base represents growth factor. Notice that in an exponential function, the variable is the exponent; this contrasts with a power function in which the variable is the base. Exponential functions change by a constant percentage for equal increments of the input value; in contrast, linear functions change by a constant amount for equal increments of the input value. This is equivalent to saying that there is a multiplicative rate of change for an exponential function in contrast to an additive rate of change for a linear function. When the base , the exponential function represents growth or increase over time; if , the exponential function represents decay or decrease over time.

To evaluate an exponential function, use the normal rules of algebra to simplify the result. For instance, given the function with . Then and . Similarly if function is defined as , then and .

The base is a special case as a base in exponential functions with many applications in the real world. The base e represents the value of as . Most scientific and graphing calculators have a special key for that can be used to evaluate exponential functions with base .

A logarithmic function is a function of the form , where , and ; this logarithmic function is read "the logarithm with base of and is equivalent to . Typically, a logarithmic function is evaluated by rewriting it in exponential form and then solving based on what is known about exponential relationships. For instance, to evaluate , rewrite the logarithm as ; so meaning that . Likewise, to evaluate rewrite in exponential form as ; because , .

There are two special logarithms that appear in many applications. Logarithms with base 10 are called common logarithms; rather than writing , common logarithms are generally written as . Natural logarithms are logarithms with base , and these logarithms appear regularly in calculus and many scientific applications. Natural logarithms are typically written as rather than as . Most scientific and graphing calculators have special keys to evaluate common logarithms (log) or natural logarithms .

Review this material in Properties of Exponential Functions and Characterisitics of Graphs of Logarithmic Functions.

7b. define the equation for exponential and logarithmic functions given a graph or data points

Describe how the equation of an exponential function can be written by knowing the initial value and growth factor.
Given a set of data points for an exponential or logarithmic function, describe how to obtain an equation for the function through those points.
Given the graph of an exponential or logarithmic function, explain how to obtain an equation for the graph.

Determining the equation of an exponential function depends on the information that is given. In some contexts, the initial value is known, corresponding to in the definition of an exponential function, and the growth factor is also known, corresponding to the base in the exponential function. Consider the following situation.

In this context, the initial value is 1200, so . A growth of per year means that the population is the base population of the base population, or times the base population; this means . Then an equation for the population after years is . After 3 years, the equation predicts a population of people.

If two data points are given, then these can be substituted into the form of an exponential function to find values for and . If one of those points corresponds to the -intercept, the work is simplified because the intercept corresponds to . For instance, consider an exponential function containing the ordered pairs and . The first ordered pair corresponds to the -intercept, so . Then substitute the second ordered pair in the equation to get ; this leads to , or because the base must be positive. Thus, the desired equation is .

If neither point corresponds to the -intercept, this process of substitution must be repeated more than once. Suppose an exponential function contains the ordered pairs and . Substituting these coordinates into the form of an exponential function yields and . The first equation leads to , which can be substituted into the second equation to obtain or ; this means that . Then the final equation is .

If a graph of an exponential function is given rather than a context or a set of ordered pairs, the techniques described above are still used. Identify two points on the graph, preferably with one of the points being the intercept. Then substitute to find and for the equation .

If the points are on a logarithmic function, similar techniques are used. A logarithmic function does not have a -intercept, unless there has been some type of transformation. However, the function does have a -intercept at ; if the -intercept is at some other point , this is a clue that the function has been shifted horizontally units from . Because is equivalent to , this relationship is used, together with knowledge of powers of common positive integers, to find a potential equation.

Consider a logarithmic function which contains the ordered pairs and . If , then these ordered pairs can be substituted into the equivalent exponential form to yield and . These two equations lead to , meaning the logarithmic equation is .

Now consider the logarithmic function whose graph is shown below. Notice that the -intercept occurs at , meaning the -intercept of the basic logarithmic function has been translated 2 units to the left. This provides a starting point of . The point also appears to be on the graph. So, substitute this point into the exponential equivalent of the logarithmic equation to obtain , so . The logarithmic function is the common logarithm function .

Review this material in Properties of Exponential Functions and Characterisitics of Graphs of Logarithmic Functions.

7c. identify properties of exponential and logarithmic equations, including asymptotes, long run, and local behavior

Identify the properties of exponential functions, including intercepts, asymptotes, and end behavior.
Identify the properties of logarithmic functions, including intercepts, asymptotes, and end behavior.
Explain how the properties of the logarithmic function are related to those of the corresponding exponential function.

The two basic exponential functions are graphed below: for and for These two functions have similar properties:

The logarithmic function is equivalent to , so the two functions are inverses of each other. This means the domain of the logarithmic function is the range of the exponential function; likewise, the range of the logarithmic function is the domain of the exponential function. Because the two functions are inverses of each, the graph of the logarithmic function is the reflection of the graph of the exponential function across the line as reviewed in objective .

Two basic logarithmic functions and are graphed below. The logarithmic function for is the inverse of the exponential function graphed above. Likewise, the logarithmic function for is the inverse of the exponential function graphed above. Both graphs have similar properties which correspond to the properties of the related exponential function. <ul> <li>Both functions are one-to-one functions.</li> <li>Neither function has a <a target="_blank" href="https://learn.saylor.org/filter/tex/displaytex.php?texexp=y" id="action_link6338bca6ca3f463" title="TeX"><img class="texrender" title="y" alt="y" src="https://learn.saylor.org/filter/tex/pix.php/415290769594460e2e485922904f345d.svg"></a><script type="math/tex">y-intercept. <li>Both functions have a <a target="_blank" href="https://learn.saylor.org/filter/tex/displaytex.php?texexp=x" id="action_link6338bca6ca3f490" title="TeX"><img class="texrender" title="x" alt="x" src="https://learn.saylor.org/filter/tex/pix.php/9dd4e461268c8034f5c8564e155c67a6.svg"></a><script type="math/tex">x

-intercept at .

Both functions contain the point .

For both functions, there is a vertical asymptote at . For function with as ; for function with as .

For both functions, the domain is ; the range is .

For function as ; for function as .

When , the function is always increasing; when , the function is always decreasing.

Review this material in Characteristics of Graphs of Exponential Functions and Characterisitics of Graphs of Logarithmic Functions.

7d. graph exponential and logarithmic equations using transformations

Describe how transformations such as vertical stretches or compressions, horizontal or vertical shifts, or vertical reflections can be used to graph exponential functions.
Compare the use of transformations when graphing exponential functions to their use when graphing logarithmic functions.

Graphing exponential and logarithmic functions using transformations involves the same techniques used in earlier chapters, particularly in objective 3c. Those same transformations apply to the concepts associated with the exponential function, such as the -intercept, horizontal asymptote, and range. Consider the function . When compared to the parent function for , the transformed function has been

For instance, consider the function . To graph, first graph the parent function (the red graph below); two points on this graph are and . Apply the transformations thinking about the order of operations. Shift the graph to the left 1 unit because (the blue graph); the points and now correspond to and . Apply a vertical stretch by a factor of 2 because (the green graph); points and correspond to and . Finally, shift the graph down 3 units because (the black graph); the points and correspond to points and on the final graph. The domain is still but the range is now ; there is a horizontal asymptote at .

Similar analysis is applied to graph a translated logarithmic function. Consider the function . When compared to the parent function for , the transformed function has been

The -intercept, domain, and equation for a vertical asymptote will be transformed in similar ways.

Consider the function . First graph the parent function the red graph below); the points and are on this graph. Apply the transformations thinking about the order of operations. Reflect the graph vertically over the -axis (the blue graph); the points and correspond to and . Because , the graph now shifts horizontally 4 units to the right (the green graph); the points and correspond to and . Finally, because , shift the graph vertically up 3 units (the black graph); the points and correspond to and on the final graph. The vertical asymptote for function is translated 4 units to the right for function and is at . The domain is now and the range is still .

Review this material in Characteristics of Graphs of Exponential Functions and Characterisitics of Graphs of Logarithmic Functions.

7e. identify the domain and range of exponential and logarithmic functions

Identify the domain and range of the basic exponential and logarithmic functions.
Explain how the domain and range of the basic exponential and logarithmic functions change when transformations are applied to the basic functions.

As described in objective 7c, the basic exponential function for has domain and range . This means that the basic logarithmic function for , and has domain and range . A transformation of the basic exponential function will not affect the domain, but the range will adjust based on any vertical shifts. A transformation of the basic logarithmic function will not affect the range, but will affect the domain because the input of a logarithmic function must be positive.

For instance, consider the exponential function . The domain is still ; however, the range will be . Observe that for all values of , meaning that .

Given the logarithmic function . The input for the logarithm must be positive, that is, ; so, the domain is . The range continues to be .

Review this material in Characteristics of Graphs of Exponential Functions and Characterisitics of Graphs of Logarithmic Functions.

7f. summarize the inverse relationship between exponential and logarithmic functions

Describe the inverse relationship between exponential and logarithmic functions.
What properties are explained by the fact that exponential and logarithmic functions are inverses of each other?

As indicated in objectives 7 a and 7c, exponential functions and logarithmic functions are inverses of each other. This relationship is used to evaluate and rewrite logarithmic functions in their equivalent exponential form. That is, for , and is equivalent to for and . Given that both functions are inverses of each other, they are both one-to-one functions. The domain and range of the exponential function are the range and domain, respectively, of the logarithmic function.

Review the material in Properties of Logarithms.