Graphs of 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 $2{,}500 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 $A=2{,}500e^{0.05t}$.

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? The graph below shows this point on the logarithmic graph.

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.

Finding the domain of a logarithmic function

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 $b>0$, $b\ne1$, where

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

In Logarithmic Functions, 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.

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 $x$:

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

Example. 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,

The domain of $f(x)=\log_2(x+3)$ is $(-3,\infty)$.

Example. What is the domain of $f(x)=\log(5-2x)$?

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

The domain of $f(x)=\log(5-2x)$ is $\left(-\infty,\tfrac{5}{2}\right)$.

Graphing logarithmic functions

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 the table below.

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

As we’d expect, the $x$- and $y$-coordinates are reversed for the inverse functions. The graph below shows the graph of $f$ and $g$.

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 $g(x)=\log_2(x)$.

The graph below 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 $e\approx2.718$.)

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

Solution. Before graphing, identify the behavior and key points for the graph.

• 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 the graph below).

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

Graphing transformations of logarithmic functions

As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function $y=\log_b(x)$ without loss of shape.

Graphing a horizontal shift of $f(x) = \log_b(x)$

When a constant $c$ is added to the input of the parent function $f(x)=\log_b(x)$, the result is a horizontal shift $c$ units in the opposite direction of the sign on $c$. To visualize horizontal shifts, we can observe the general graph of the parent function $f(x)=\log_b(x)$ for $c>0$ alongside the shift left, $g(x)=\log_b(x+c)$, and the shift right, $h(x)=\log_b(x-c)$. See the panels below, drawn with $b=2$ and $c=2$.

Example. Sketch the horizontal shift $f(x)=\log_3(x-2)$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Solution. Since the function is $f(x)=\log_3(x-2)$, we notice $x+(-2)=x-2$.

Thus $c=-2$, so $c<0$. This means we will shift the function $f(x)=\log_3(x)$ right 2 units.

The vertical asymptote is $x=-(-2)$ or $x=2$.

Consider the three key points from the parent function, $\left(\tfrac{1}{3},-1\right)$, $(1,0)$, and $(3,1)$.

The new coordinates are found by adding 2 to the $x$ coordinates.

Label the points $\left(\tfrac{7}{3},-1\right)$, $(3,0)$, and $(5,1)$.

The domain is $(2,\infty)$, the range is $(-\infty,\infty)$, and the vertical asymptote is $x=2$.

Graphing a vertical shift of $f(x) = \log_b(x)$

When a constant $d$ is added to the parent function $f(x)=\log_b(x)$, the result is a vertical shift $d$ units in the direction of the sign on $d$. To visualize vertical shifts, we can observe the general graph of the parent function $f(x)=\log_b(x)$ alongside the shift up, $g(x)=\log_b(x)+d$, and the shift down, $h(x)=\log_b(x)-d$. See the panels below, drawn with $b=2$ and $d=2$.

Example. Sketch a graph of $f(x)=\log_3(x)-2$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Solution. Since the function is $f(x)=\log_3(x)-2$, we will notice $d=-2$. Thus $d<0$.

This means we will shift the function $f(x)=\log_3(x)$ down 2 units.

The vertical asymptote is $x=0$.

Consider the three key points from the parent function, $\left(\tfrac{1}{3},-1\right)$, $(1,0)$, and $(3,1)$.

The new coordinates are found by subtracting 2 from the $y$ coordinates.

Label the points $\left(\tfrac{1}{3},-3\right)$, $(1,-2)$, and $(3,-1)$.

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

Graphing stretches and compressions of $f(x) = \log_b(x)$

When the parent function $f(x)=\log_b(x)$ is multiplied by a constant $a>0$, the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set $a>1$ and observe the general graph of the parent function $f(x)=\log_b(x)$ alongside the vertical stretch, $g(x)=a\log_b(x)$, and the vertical compression, $h(x)=\tfrac{1}{a}\log_b(x)$. See the panels below, drawn with $b=2$ and $a=2$.

Example. Sketch a graph of $f(x)=2\log_4(x)$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Solution. Since the function is $f(x)=2\log_4(x)$, we will notice $a=2$.

This means we will stretch the function $f(x)=\log_4(x)$ by a factor of 2.

The vertical asymptote is $x=0$.

Consider the three key points from the parent function, $\left(\tfrac{1}{4},-1\right)$, $(1,0)$, and $(4,1)$.

The new coordinates are found by multiplying the $y$ coordinates by 2.

Label the points $\left(\tfrac{1}{4},-2\right)$, $(1,0)$, and $(4,2)$.

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

Example. Sketch a graph of $f(x)=5\log(x+2)$. State the domain, range, and asymptote.

Solution. Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in the graph below. The vertical asymptote will be shifted to $x=-2$. The $x$-intercept will be $(-1,0)$. The domain will be $(-2,\infty)$. Two points will help give the shape of the graph: $(-1,0)$ and $(8,5)$. We chose $x=8$ as the $x$-coordinate of one point to graph because when $x=8$, $x+2=10$, the base of the common logarithm.

The domain is $(-2,\infty)$, the range is $(-\infty,\infty)$, and the vertical asymptote is $x=-2$.

Graphing reflections of $f(x) = \log_b(x)$

When the parent function $f(x)=\log_b(x)$ is multiplied by $-1$, the result is a reflection about the $x$-axis. When the input is multiplied by $-1$, the result is a reflection about the $y$-axis. To visualize reflections, we restrict $b>1$, and observe the general graph of the parent function $f(x)=\log_b(x)$ alongside the reflection about the $x$-axis, $g(x)=-\log_b(x)$, and the reflection about the $y$-axis, $h(x)=\log_b(-x)$. Both are drawn below with $b=2$.

Example. Sketch a graph of $f(x)=\log(-x)$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Solution. Before graphing $f(x)=\log(-x)$, identify the behavior and key points for the graph.

• Since $b=10$ is greater than one, we know that the parent function is increasing. Since the input value is multiplied by $-1$, $f$ is a reflection of the parent graph about the $y$-axis. Thus, $f(x)=\log(-x)$ will be decreasing as $x$ moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote $x=0$.
• The $x$-intercept is $(-1,0)$.
• We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.

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

Example. Solve $4\ln(x)+1=-2\ln(x-1)$ graphically. Round to the nearest thousandth.

Solution. Press [Y=] and enter $4\ln(x)+1$ next to Y1=. Then enter $-2\ln(x-1)$ next to Y2=. For a window, use the values 0 to 5 for $x$ and $-10$ to $10$ for $y$. Press [GRAPH]. The graphs should intersect somewhere a little to right of $x=1$.

For a better approximation, press [2ND] then [CALC]. Select [5: intersect] and press [ENTER] three times. The $x$-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for Guess?.) So, to the nearest thousandth, $x\approx1.339$.

Summarizing translations of the logarithmic function

Now that we have worked with each type of translation for the logarithmic function, we can summarize each in the table below to arrive at the general equation for translating logarithmic functions.

Example. What is the vertical asymptote of $f(x)=-2\log_3(x+4)+5$?

Solution. The vertical asymptote is at $x=-4$.

Analysis. The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to $x=-4$.

Example. Find a possible equation for the common logarithmic function graphed below.

Solution. This graph has a vertical asymptote at $x=-2$ and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:

It appears the graph passes through the points $(-1,1)$ and $(2,-1)$. Substituting $(-1,1)$,

Next, substituting in $(2,-1)$,

This gives us the equation $f(x)=-\tfrac{2}{\log(4)}\log(x+2)+1$.

Analysis. We can verify this answer by comparing the function values in the table below with the points on the graph above.

Key equations

Key concepts

• To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for $x$.
• The graph of the parent function $f(x)=\log_b(x)$ has an $x$-intercept at $(1,0)$, domain $(0,\infty)$, range $(-\infty,\infty)$, vertical asymptote $x=0$, and
  • if $b>1$, the function is increasing.
  • if $0<b<1$, the function is decreasing.
• The equation $f(x)=\log_b(x+c)$ shifts the parent function $y=\log_b(x)$ horizontally
  • left $c$ units if $c>0$.
  • right $c$ units if $c<0$.
• The equation $f(x)=\log_b(x)+d$ shifts the parent function $y=\log_b(x)$ vertically
  • up $d$ units if $d>0$.
  • down $d$ units if $d<0$.
• For any constant $a>0$, the equation $f(x)=a\log_b(x)$
  • stretches the parent function $y=\log_b(x)$ vertically by a factor of $a$ if $|a|>1$.
  • compresses the parent function $y=\log_b(x)$ vertically by a factor of $a$ if $|a|<1$.
• When the parent function $y=\log_b(x)$ is multiplied by $-1$, the result is a reflection about the $x$-axis. When the input is multiplied by $-1$, the result is a reflection about the $y$-axis.
  • The equation $f(x)=-\log_b(x)$ represents a reflection of the parent function about the $x$-axis.
  • The equation $f(x)=\log_b(-x)$ represents a reflection of the parent function about the $y$-axis.
• A graphing calculator may be used to approximate solutions to some logarithmic equations.
• All translations of the logarithmic function can be summarized by the general equation $f(x)=a\log_b(x+c)+d$.
• Given an equation with the general form $f(x)=a\log_b(x+c)+d$, we can identify the vertical asymptote $x=-c$ for the transformation.
• Using the general equation $f(x)=a\log_b(x+c)+d$, we can write the equation of a logarithmic function given its graph.

Practice

Identify the domain of a logarithmic function

Graph logarithmic functions

The five curves below all share the point $(1,0)$ and are labeled A through E.

This section is adapted from Precalculus 2e, Section 4.4: Graphs of Logarithmic Functions by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated every graph as an accessible spec-first SVG built from its exact equation — the investment logarithmic model $t=20\ln(A)-20\ln(2{,}500)$ with its labeled “balance reaches $5{,}000 near year 14” point; the $f(x)=2^x$/$g(x)=\log_2(x)$ reflection pair about $y=x$; the two-panel parent-function characteristics schematic; the three-curve base-comparison graph $\log_2(x)$, $\ln(x)$, and $\log(x)$; the $f(x)=\log_5(x)$ worked example; the two-panel horizontal-shift, vertical-shift, stretch/compression, and reflection schematics (recreated with concrete representative values $b=2$ and $c=2$, $d=2$, or $a=2$ in place of the source’s symbolic point labels, since the figure engine draws real analytic curves rather than schematic ones); the horizontal-shift, vertical-shift, stretch, and combined shift-and-stretch worked examples; the reflection-about-the-$y$-axis panel and the $f(x)=\log(-x)$ worked example, both of which need $x<h$ on their reflected branch — outside the analytic log primitive’s domain — and so are rendered as a dense reflected log curve primitive (added to graph-core.mjs in this pass, which previously could only open rightward from a vertical asymptote); the vertical-asymptote and equation-from-a-graph worked examples; and the five-curve base-comparison figure and the equation-from-a-graph item used in the closing Practice block. Converted the two-column “graph a reflection” How To table into two separate How To callouts, one per reflection axis, matching this book’s callout convention. Presented the four data tables (the $2^x$/$\log_2(y)$ correspondence, the $f$/$g$ ordered-pair correspondence, and the two-part function-value check table, the latter combined into a single ten-column table since its source split was only print pagination) as Markdown tables. Omitted the “Access these online resources” media links. Omitted the corequisite-skills “Objective 1”/“Objective 2” review appendix that precedes this section’s actual content in the pinned CNXML module, since it does not appear in the printed textbook (confirmed against the source PDF) and duplicates material this section itself develops. Converted the ten retained “Try It” checks into interactive fill-ins with instant feedback, several split into a domain question and a vertical-asymptote or key-point question when the original asked for multiple facts at once; every exponential and logarithmic curve in this section is graded through a static figure plus a fill-in or multiple-choice question, except the vertical-shift Try It $f(x)=\log_2(x)+2$, which is additionally graded as a plot-the-points graph exercise at five named inputs ($x=\tfrac{1}{2}$ through $8$, whose outputs land on a half-unit grid) — the plotted points are graded, never the curve through them. And adapted five selected end-of-section exercises — two domain evaluations, one vertical-asymptote evaluation, a graph-matching pair converted to graph-mode multiple choice, and one equation-from-a-graph item — into interactive components in a closing Practice block, one group per objective.