Characterisitics of Graphs of Logarithmic Functions

Learning Objectives

In this section, you will:

• Identify the domain of a logarithmic function.
• Graph logarithmic functions.

In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect.

To illustrate, suppose we invest $$2500$ in an account that offers an annual interest rate of $5 \%$, compounded continuously. We already know that the balance in our account for any year $t$ can be found with the equation $A=2500e^{0.05t}$.

But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? Figure 1 shows this point on the logarithmic graph.

Figure 1

In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.

Source: Rice University, https://openstax.org/books/college-algebra/pages/6-4-graphs-of-logarithmic-functions
This work is licensed under a Creative Commons Attribution 4.0 License.

Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.

Recall that the exponential function is defined as $y=b^x$ for any real number $x$ and constant $b > 0, b \neq 1$, where

• The domain of $y$ is $(−\infty, \infty)$.
• The range of $y$ is $(0, \infty)$.

In the last section we learned that the logarithmic function $y=log_b(x)$ is the inverse of the exponential function $y=b^x$. So, as inverse functions:

• The domain of $y=log_b(x)$ is the range of $y=b^x : (0, \infty)$.
• The range of $y=log_b(x)$ is the domain of $y=b^x : (−\infty, \infty)$.

Transformations of the parent function $y=log_b(x)$ behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations – shifts, stretches, compressions, and reflections – to the parent function without loss of shape.

In Graphs of Exponential Functions we saw that certain transformations can change the range of $y=b^x$. Similarly, applying transformations to the parent function $y=log_b(x)$ can change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. That is, the argument of the logarithmic function must be greater than zero.

For example, consider $f(x)=log_4(2x−3)$. This function is defined for any values of $x$ such that the argument, in this case $2x−3$, is greater than zero. To find the domain, we set up an inequality and solve for $x$:

$2x−3 > 0 \quad \text{Show the argument greater than zero.}$

$2x > 3 \quad \text{Add} 3.$

$x > 1.5 \quad \text{Divide by} 2.$

In interval notation, the domain of $f(x)=log_4(2x−3)$ is $(1.5, \infty)$.

HOW TO

Given a logarithmic function, identify the domain.

1. Set up an inequality showing the argument greater than zero.
2. Solve for  $x$.
3. Write the domain in interval notation.

EXAMPLE 1

Identifying the Domain of a Logarithmic Shift

What is the domain of $f(x)=log_2(x+3)?$

Solution

The logarithmic function is defined only when the input is positive, so this function is defined when $x+3 > 0$. Solving this inequality,

$\begin{aligned} &x+3>0 & & \text { The input must be positive. } \\ &x > −3 & & \text { Subtract } 3. \end{aligned}$

TRY IT #1

What is the domain of $f(x)=log_5(x−2)+1?$

EXAMPLE 2

Identifying the Domain of a Logarithmic Shift and Reflection

Solution

The logarithmic function is defined only when the input is positive, so this function is defined when $5–2x > 0$. Solving this inequality,

$5-2 x > 0 \quad \text{The input must be positive.}$

$-2 x > -5 \quad \text { Subtract } 5.$

$x < \frac{5}{2} \quad \text { Divide by } -2 \text { and switch the inequality. }$

The domain of $f(x)=log(5−2x)$ is $(– \infty, \frac{5}{2})$.

TRY IT #2

What is the domain of $f(x)=log(x−5)+2?$

Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function $y=log_b(x)$ along with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the parent function $y=log_b(x)$. Because every logarithmic function of this form is the inverse of an exponential function with the form $y=b^x$, their graphs will be reflections of each other across the line $y=x$. To illustrate this, we can observe the relationship between the input and output values of $y=2^x$ and its equivalent $x=log_2(y)$ in Table 1.

Table 1

Using the inputs and outputs from Table 1, we can build another table to observe the relationship between points on the graphs of the inverse functions $f(x)=2^x$ and $g(x)=log_2(x)$. See Table 2.

Table 2

Figure 2 Notice that the graphs of $f(x)=2^x$ and $g(x)=log_2(x)$ are reflections about the line $y=x$.

Observe the following from the graph:

• $f(x)=2^x$ has a $y$-intercept at $(0,1)$ and $g(x)=log_2(x)$ has an $x$- intercept at $(1,0)$.
• The domain of $f(x)=2^x, (− \infty, \infty)$, is the same as the range of $g(x)=log_2(x)$.
• The range of $f(x)=2^x, (0, \infty)$, is the same as the domain of $g(x)=log_2(x)$.

CHARACTERISTICS OF THE GRAPH OF THE PARENT FUNCTION, $f(x)=log_b(x)$:

For any real number $x$ and constant $b > 0, b \neq 1$, we can see the following characteristics in the graph of $f(x)=log_b(x)$:

• one-to-one function
• vertical asymptote: $x=0$
• domain: $(0, \infty)$
• range: $(−\infty, \infty)$
• $x$-intercept: $(1,0)$ and key point $(b,1)$
• $y$-intercept: none
• increasing if $b > 1$
• decreasing if $0 < b < 1$

See Figure 3.

Figure 3

Figure 4 shows how changing the base $b$ in $f(x)=log_b(x)$ can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function $\ln(x)$ has base $e \approx 2.718$.)

Figure 4 The graphs of three logarithmic functions with different bases, all greater than 1.

HOW TO

Given a logarithmic function with the form $f(x)=log_b(x)$, graph the function.

1. Draw and label the vertical asymptote, $x=0$.
2. Plot the $x$-intercept, $(1,0)$.
3. Plot the key point $(b,1)$.
4. Draw a smooth curve through the points.
5. State the domain, $(0, \infty)$, the range, $(−\infty, \infty)$, and the vertical asymptote, $x=0$.

EXAMPLE 3

Graphing a Logarithmic Function with the $f(x) = log_b(x)$.

Graph $f(x)=log_5(x)$. State the domain, range, and asymptote.

Solution

• Since $b=5$ is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote $x=0$, and the right tail will increase slowly without bound.
• The $x$-intercept is $(1,0)$.
• The key point $(5,1)$ is on the graph.
• We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see Figure 5).

Figure 5

The domain is $(0, \infty)$, the range is $(−\infty, \infty)$, and the vertical asymptote is $x=0$.

TRY IT #3

Graph $f(x)=log_{\frac{1}{5}}(x)$. State the domain, range, and asymptote.