MA001: College Algebra

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 $e$.
• 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 $f(x)=a b^{x}$, where $a$ and $b$ are positive real numbers and $b \neq 1$; the coefficient a represents the starting value and the base $b$ 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 $b > 1$, the exponential function represents growth or increase over time; if $0 < b < 1$, 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 $f$ with $f(x)=3 \cdot 4^{x}$. Then $f(2)=3 \cdot 4^{2}=3 \cdot 16=48$ and $f(-2)=3 \cdot 4^{-2}=3 \cdot \frac{1}{16}=\frac{3}{16}$. Similarly if function $g$ is defined as $g(x)=\left(\frac{1}{4}\right)^{x}$, then $g(2)=\left(\frac{1}{4}\right)^{2}=\frac{1}{16}$ and $g(-2)=\left(\frac{1}{4}\right)^{-2}=16$.

The base $e$ is a special case as a base in exponential functions with many applications in the real world. The base e represents the value of $\left(1+\frac{1}{n}\right)^{n}$ as $n \rightarrow \infty; e \approx 2.71828$. Most scientific and graphing calculators have a special key for $e$ that can be used to evaluate exponential functions with base $e$.

A logarithmic function is a function of the form $y=\log _{b} x$, where $x > 0, b > 0$, and $b \neq 1$; this logarithmic function is read "the logarithm with base $b$ of $x^{\prime \prime}$ and is equivalent to $b^{y}=x$. 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 $y=\log _{3} 81$, rewrite the logarithm as $3^{y}=81$; so $y=4$ meaning that $\log _{3} 81=4$. Likewise, to evaluate $y=\log _{5} \frac{1}{125}$ rewrite in exponential form as $5^{y}=\frac{1}{125}$; because $y=-3$, $\log _{5} \frac{1}{125}=-3$.

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

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 $a$ in the definition of an exponential function, and the growth factor is also known, corresponding to the base $b$ in the exponential function. Consider the following situation.

• The population of a rural area is 1200 people. If the population is expected to grow by $2 \%$ a year, what equation describes the population after $t$ years?

In this context, the initial value is 1200, so $a=1200$. A growth of $2 \%$ per year means that the population is the base population $+2 \%$ of the base population, or $1.02$ times the base population; this means $b=1.02$. Then an equation for the population after $t /$ years is $f(t)=1200 \cdot 1.02^{t}$. After 3 years, the equation predicts a population of $1200 \cdot 1.02^{3} \approx 1273$ people.

If two data points are given, then these can be substituted into the form of an exponential function to find values for $a$ and $b$. If one of those points corresponds to the $y$-intercept, the work is simplified because the $y$ intercept corresponds to $a$. For instance, consider an exponential function containing the ordered pairs $(0,7)$ and $(2,63)$. The first ordered pair corresponds to the $y$-intercept, so $a=7$. Then substitute the second ordered pair in the equation $f(x)=7 b^{x}$ to get $63=7 b^{2}$; this leads to $b^{2}=9$, or $b=3$ because the base must be positive. Thus, the desired equation is $f(x)=7 \cdot 3^{x}$.

If neither point corresponds to the $y$-intercept, this process of substitution must be repeated more than once. Suppose an exponential function contains the ordered pairs $(1,2)$ and $\left(5, \frac{1}{8}\right)$. Substituting these coordinates into the form of an exponential function yields $2=a b^{1}$ and $\frac{1}{8}=a b^{5}$. The first equation leads to $\frac{2}{a}=b$, which can be substituted into the second equation to obtain $\frac{1}{8}=a \cdot\left(\frac{2}{a}\right)^{5}$ or $a=4$; this means that $b=\frac{1}{2}$. Then the final equation is $g(x)=4 \cdot\left(\frac{1}{2}\right)^{x}$.

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 $y$ intercept. Then substitute to find $a$ and $b$ for the equation $f(x)=a b^{x}$.

If the points are on a logarithmic function, similar techniques are used. A logarithmic function does not have a $y$-intercept, unless there has been some type of transformation. However, the function does have a $x$-intercept at $(1,0)$; if the $x$-intercept is at some other point $(h, 0)$, this is a clue that the function has been shifted horizontally $h$ units from $(1,0)$. Because $y=\log _{b} x$ is equivalent to $b^{y}=x$, 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 $(36,2)$ and $(216,3)$. If $y=\log _{b} x$, then these ordered pairs can be substituted into the equivalent exponential form to yield $36=b^{2}$ and $216=b^{3}$. These two equations lead to $b=6$, meaning the logarithmic equation is $y=\log _{6} x$.

Now consider the logarithmic function whose graph is shown below. Notice that the $x$-intercept occurs at $(-1,0)$, meaning the $x$-intercept of the basic logarithmic function $(1,0)$ has been translated 2 units to the left. This provides a starting point of $y=\log _{b}(x+2)$. The point $(8,1)$ also appears to be on the graph. So, substitute this point into the exponential equivalent of the logarithmic equation to obtain $b^{1}=8+2$, so $b=10$. The logarithmic function is the common logarithm function $f(x)=\log (x+2)$.

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: $f(x)=b^{x}$ for $b > 1$ and $g(x)=b^{x}$ for $0 < b < 1.$ These two functions have similar properties:

• Both functions are one-to-one functions.
• For both functions, the $y$-intercept is at $(0, \, \mid 1)$.
• Neither function has a $x$-intercept.
• Both functions contain the point $(1, \, b)$.
• For both functions, there is a horizontal asymptote at $y=0$. For function $f$ with $b > 1, f(x) \rightarrow 0$ as $x \rightarrow$ $-\infty$; for function $g$ with $0 < b < 1, g(x) \rightarrow 0$ as $x \rightarrow \infty$.
• For both functions, the domain is $(-\infty, \infty)$; the range is $(0, \, \infty)$.
• For function $f$ with $b > 1, f(x) \rightarrow \infty$ as $x \rightarrow \infty$; for function $g$ with $0 < b < 1, g(x) \rightarrow 0$ as $x \rightarrow \infty$.
• For $b > 1$, the function is always increasing; when $0 < b < 1$, the function is always decreasing.

The logarithmic function $y=f(x)=\log _{b} x$ is equivalent to $b^{y}=x$, 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 $y=x$ as reviewed in objective $3 \mathrm{~g}$.

Two basic logarithmic functions $p$ and $q$ are graphed below. The logarithmic function $y=p(x)=\log _{b} x$ for $b > 1$ is the inverse of the exponential function $y=f(x)$ graphed above. Likewise, the logarithmic function $y=$ $q(x)=\log _{b} x$ for $0 < b < 1$ is the inverse of the exponential function $y=g(x)$ graphed above. Both graphs have similar properties which correspond to the properties of the related exponential function.

• Both functions are one-to-one functions.
• Neither function has a $y$-intercept.
• Both functions have a $x$-intercept at $(1,0)$.
• Both functions contain the point $(b, 1)$.
• For both functions, there is a vertical asymptote at $x=0$. For function $p$ with $b > 1, p(x) \rightarrow-\infty$ as $x \rightarrow$ $0^{+}$; for function $q$ with $0 < b < 1, q(x) \rightarrow \infty$ as $x \rightarrow 0^{+}$.
• For both functions, the domain is $(0,  \,  \infty)$; the range is $(-\infty,  \,  \infty)$.
• For function $p,  \,  p(x) \rightarrow \infty$ as $x \rightarrow \infty$; for function $q,  \,  q(x) \rightarrow-\infty$ as $x \rightarrow \infty$.
• When $b > 1$, the function is always increasing; when $0 < b < 1$, 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 $y$-intercept, horizontal asymptote, and range. Consider the function $f(x)=a b^{x+c}+d$. When compared to the parent function $f(x)=b^{x}$ for $b > 1$, the transformed function has been

• Shifted horizontally by $c$ units (to the left if $c > 0$, to the right if $c < 0$).
• Stretched vertically by $a$ if $|a| > 1$ or compressed vertically if $0 < |a| < 1$.
• Shifted vertically by $d$ units (up if $d > 0$, down if $d < 0$).
• Reflected vertically across the $x$-axis if $a < 0$.

For instance, consider the function $y=f(x)=2 \cdot 4^{x+1}-3$. To graph, first graph the parent function $g(x)=4^{x}$ (the red graph below); two points on this graph are $(0,1)$ and $(1,4)$. Apply the transformations thinking about the order of operations. Shift the graph to the left 1 unit because $c=1$ (the blue graph); the points $(0,1)$ and $(1,  \,  4)$ now correspond to $(-1,1)$ and $(0,4)$. Apply a vertical stretch by a factor of 2 because $a=2$ (the green graph); points $(-1,1)$ and $(0,4)$ correspond to $(-1, \, 2)$ and $(0.8)$. Finally, shift the graph down 3 units because $d=-3$ (the black graph); the points $(-1,2)$ and $(0,8)$ correspond to points $(-1, \, -1)$ and $(0,5)$ on the final graph. The domain is still $(-\infty, \infty)$ but the range is now $(-3, \,  \infty)$; there is a horizontal asymptote at $y=-3$.

Similar analysis is applied to graph a translated logarithmic function. Consider the function $f(x)=a l o g_{b}(x+$ $c)+d$. When compared to the parent function $f(x)=\log _{b} x$ for $b > 1$, the transformed function has been

• Shifted vertically by $d$ units (up if $d > 0$, down if $d < 0$).
• Shifted horizontally by $c$ units (to the left if $c > 0$, to the right if $c < 0$).
• Stretched vertically by $a$ if $|a| > 1$ or compressed vertically if $0 < |a| < 1$.
• Reflected horizontally about the $x$-axis if $a < 0$.

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

Consider the function $y=f(x)=-\log (x-4)+3$. First graph the parent function $g(x)=\log x($ the red graph below); the points $(1, \, 0)$ and $(10, \, 1)$ are on this graph. Apply the transformations thinking about the order of operations. Reflect the graph vertically over the $x$-axis (the blue graph); the points $(1,0)$ and $(10,1)$ correspond to $(1, \, 0)$ and $(10, \, -1)$. Because $c=-4$, the graph now shifts horizontally 4 units to the right (the green graph); the points $(1, \, 0)$ and $(10, \, -1)$ correspond to $(5, \, 0)$ and $(14, \, -1)$. Finally, because $d=3$, shift the graph vertically up 3 units (the black graph); the points $(5, \, 0)$ and $(14, \, -1)$ correspond to $(5, \, 3)$ and $(14, \, 2)$ on the final graph. The vertical asymptote for function $g$ is translated 4 units to the right for function $f$ and is at $x=4$. The domain is now $(4,  \, \infty)$ and the range is still $(-\infty, \infty)$.

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 $f(x)=b^{x}$ for $b > 0, b \neq 1$ has domain $(-\infty, \infty)$ and range $(0, \infty)$. This means that the basic logarithmic function $g(x)=\log _{b} x$ for $x > 0, b > 0$, and $b \neq 1$ has domain $(0, \infty)$ and range $(-\infty, \infty)$. 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 $y=5 \cdot 7^{x+4}-6$. The domain is still $(-\infty, \infty)$; however, the range will be $(-6, \infty)$. Observe that $5 \cdot 7^{x+4} > 0$ for all values of $x$, meaning that $y > -6$.

Given the logarithmic function $y=6 \log _{5}(x-8)+4$. The input for the logarithm must be positive, that is, $x-8 > 0$; so, the domain is $(8, \infty)$. The range continues to be $(-\infty, \infty)$.

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, $y=\log _{b} x$ for $x > 0, b > 0$, and $b \neq 1$ is equivalent to $b^{y}=x$ for $b > 0$ and $b \neq 1$. 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.

Unit 7 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• base of an exponential function
• common logarithm
• $e$
• exponential function
• logarithmic function
• natural logarithm